p-adic Numbers, From the Ground Up

A self-contained course built for the Map of Math project — the second strand, and it plugs directly into the first (elliptic-curves-course.md) at several points, which is called out explicitly where it happens.

How to use this document

Same shape as the elliptic curves course, deliberately: motivation first, then the real history (so the ideas arrive in the order humans needed them), then a foundations section built toward exactly one destination — the notion of completing a field in a new way — then eight coursework modules (exposition, one fully worked example, a graded problem set), a full answer key, an organized resource list, and finally a connection graph showing where this sits in your wider map and what to open next. Work the problems before reading the answers.

One framing note up front, because it’s the whole point of this course: p-adic numbers are usually taught as a strange add-on, a curiosity bolted onto “real” mathematics. That’s backwards, and it’s a historical accident of which completion of $\mathbb{Q}$ your teachers happened to grow up with. By the end of §1–§2 the goal is for $\mathbb{Q}_p$ to feel exactly as natural as $\mathbb{R}$ — two members of the same family, not one normal object and one exotic one.

1. Motivation — why this object, specifically

Here’s a question nobody asks you in school, because by the time you meet $\mathbb{R}$ it’s already been handed to you as the completion of $\mathbb{Q}$, full stop: why should the ordinary absolute value $|x-y|$ be the only reasonable way to measure how “close” two rational numbers are?

Ordinary distance says $1000000$ and $1000001$ are far apart (they differ by 1… wait, they’re close) and $1$ and $1000000$ are far apart. It’s built entirely around size. But there’s a completely different, equally legitimate notion of closeness hiding in the rationals, one built around divisibility instead of size: fix a prime $p$, and declare two rational numbers close if their difference is divisible by a large power of $p$. Under this rule, $1$ and $1{,}000{,}001$ are extremely close 5-adically (their difference $1{,}000{,}000 = 2^6 \cdot 5^6$ is divisible by $5^6$), even though they’re about as far apart as two numbers can be in the ordinary sense. This isn’t a trick or an analogy — it satisfies every axiom a distance function needs (Module 1 makes this precise), and when you complete $\mathbb{Q}$ with respect to it, exactly the way Cauchy sequences complete $\mathbb{Q}$ into $\mathbb{R}$, you get a genuinely new, genuinely complete field: $\mathbb{Q}_p$, the $p$-adic numbers.

The hindsight-obvious version of this, which took mathematics about twenty years to formalize after the numbers themselves were constructed (§2 has the real timeline): Ostrowski’s theorem (1916) says every possible notion of absolute value on $\mathbb{Q}$ — up to the natural equivalence that doesn’t change which sequences converge — is either the usual one, or one of these $p$-adic ones, one for each prime. There is no third kind, and there never will be another one discovered later; the list is closed and complete. That means the correct picture of “completing the rationals” was never a single act producing $\mathbb{R}$. It was always a family of acts, indexed by the places of $\mathbb{Q}$ — one place for every prime $p$, plus one more place (traditionally called the “infinite place” or $\infty$) for the usual absolute value. $\mathbb{R}$ is the completion at $\infty$. $\mathbb{Q}_2, \mathbb{Q}_3, \mathbb{Q}_5, \mathbb{Q}_7, \dots$ are the completions at each finite prime. They are all, formally, equally valid completions of the same field. $\mathbb{R}$ just happens to be the one you meet first, because human beings evolved to measure size, not divisibility — an accident of biology, not of mathematics.

There’s a second hindsight lens that makes the construction feel inevitable rather than clever, and it’s the one that actually motivated Hensel in 1897 (§2): the function field analogy. Consider the ring of polynomials $K[x]$ instead of $\mathbb{Z}$, and the field of rational functions $K(x)$ instead of $\mathbb{Q}$. If you want to study a rational function $f(x)$ near a specific point $x=a$, the natural tool is its Taylor/Laurent expansion around that point: $f(x) = \sum_{n \ge N} c_n (x-a)^n$. This expansion converges (formally, or in the usual sense if $K=\mathbb{C}$) precisely near $a$, and it lets you do local analysis — differentiate, find roots via Newton’s method, study behavior order-of-vanishing by order-of-vanishing — one point at a time, even though the global function $f$ might be complicated everywhere else. Dedekind and Weber had shown by 1882 that the arithmetic of $K[x]$ (irreducible polynomials, i.e. points on a curve, playing the role of primes) is a near-perfect mirror of the arithmetic of $\mathbb{Z}$ (primes). Hensel’s insight was to push the analogy one step further: if $\mathbb{Z}$’s primes behave like $K[x]$’s points, then number theory ought to have its own analogue of “expanding a function in a Taylor series near a point” — expanding an integer or rational number in a series in powers of a prime $p$, the number-theoretic analogue of $(x-a)$. That expansion is exactly a $p$-adic number: $a_0 + a_1 p + a_2 p^2 + \cdots$, digits $a_i \in \{0, \dots, p-1\}$, extending infinitely — not off to the right of a decimal point the way real decimals do, but off to the left, as coefficients of higher and higher powers of $p$. Once you see it this way, $p$-adic numbers stop looking like a bizarre completion trick and start looking like the obvious, overdue transplant of ordinary local analysis (Taylor series near a point) into arithmetic (power series near a prime).

Four independent reasons this deserves a deep, dedicated thread in your map, mirroring the structure that made elliptic curves worth one:

It corrects a false sense that $\mathbb{R}$ is special. Once Ostrowski’s theorem is in view, “the real numbers” is revealed as one entry in a complete, closed list of equally legitimate completions of $\mathbb{Q}$. This single reframing changes how you read almost all of classical number theory in hindsight — theorems that look like isolated facts about $\mathbb{Z}$ or $\mathbb{Q}$ are frequently theorems about all the places at once, with $\mathbb{R}$ contributing one condition and each $\mathbb{Q}_p$ contributing another.

It gives you a genuinely different geometry to think in, and it’s fully rigorous. The $p$-adic distance satisfies a stronger triangle inequality than the usual one (the ultrametric inequality, Module 1), and the consequences are wonderfully strange: every triangle is isosceles, every point inside a ball is equally entitled to be called its center, and every ball is simultaneously open and closed. This isn’t pathology — it’s a clean, internally consistent alternate geometry, and a full theory of analysis (convergence, power series, an exponential and logarithm, even integration) can be built on top of it, often with simpler convergence criteria than in $\mathbb{R}$ (Module 6).

Local information determines global information — sometimes. The local-global principle, first proved for quadratic forms by Hasse and Minkowski in the early 1920s, says: a quadratic equation has a rational solution if and only if it has a solution in $\mathbb{R}$ and a solution in every single $\mathbb{Q}_p$. Checking infinitely many local conditions (one per prime) turns out to be a finite, mechanical computation, and it completely settles a global existence question. This “check everywhere locally, deduce it globally” philosophy — and the famous cases where it fails (Module 7) — is one of the central organizing ideas of modern number theory, including the Selmer group machinery used to bound ranks of elliptic curves in your other course document.

It is quietly load-bearing under some of the deepest 20th-century mathematics you’ve already touched. Andrew Wiles’s proof of Fermat’s Last Theorem (your elliptic curves course, Module 6–7) works by deforming Galois representations valued in $\mathrm{GL}_2(\mathbb{Z}_p)$ — the coefficient ring for the whole argument is a ring of $p$-adic numbers. Iwasawa theory, $p$-adic $L$-functions, and $p$-adic Hodge theory — some of the most active machinery in current arithmetic geometry — are built entirely on the objects in this course. You cannot go much further into modern number theory without them.

Keep this as the throughline: everything below is one idea — that “distance” on $\mathbb{Q}$ is not unique, and every legitimate alternative gives you a new, complete, useful number system — examined from every angle mathematicians have found useful (algebraic, topological, analytic, arithmetic) over about 130 years.

2. History and intellectual lineage

As with the elliptic curves course: textbooks present this in logical order (axioms → construction → theorems), which is not the order it was actually discovered in. Here is the real order.

Two threads worth sitting with, the same way the elliptic curves history rewarded it: first, notice that $p$-adic numbers were not discovered by someone trying to build a new number system for its own sake — Hensel was trying to import a well-understood tool (local power series expansions) from function theory into number theory, and the “new number system” was a side effect of doing that rigorously. That is a recurring shape in mathematics: transplant a working method from one domain into a structurally analogous domain, and often the destination domain didn’t even have the right objects yet for the method to act on — you have to build them first. Second, notice the same long-incubation-then-surprising-collapse pattern as the Taniyama–Shimura story: a construction from 1897 (Hensel) sits as a relatively niche tool for nearly a century before becoming indispensable — not optional, indispensable — to resolving a 350-year-old problem (Wiles, 1995) that has nothing to do with power series on its face. The p-adic numbers and the elliptic curves course are not two separate stories; by 1994 they are the same story.

Sources consulted for this section: Fernando Q. Gouvêa, “Hensel’s p-adic Numbers: Early History”, Kurt Hensel — MacTutor History of Mathematics, P-adic number — Encyclopedia of Mathematics, P-adic valuation — Wikipedia, P-adic analysis — Wikipedia, An introduction to p-adic L-functions, arXiv:2309.15692.

3. Foundations, built from scratch toward completion

The elliptic curves course rebuilt groups/rings/fields from scratch en route to the group law. This course needs a different kind of scaffolding: not new algebraic structures, but a new metric structure on a field you already know everything about algebraically ($\mathbb{Q}$). Everything here points at one destination: understanding, precisely, what it means to complete $\mathbb{Q}$ “at a prime” instead of “at infinity.”

3.1 Absolute values on a field

An absolute value on a field $K$ is a function $|\cdot|: K \to \mathbb{R}_{\ge 0}$ satisfying, for all $x, y \in K$:

• $|x| = 0 \iff x = 0$,
• $|xy| = |x|\,|y|$ (multiplicativity),
• $|x + y| \le |x| + |y|$ (triangle inequality).

The familiar example on $\mathbb{Q}$ is $|x|_\infty$, ordinary absolute value. There’s also always the trivial absolute value ($|0|=0$, $|x|=1$ otherwise), which is excluded from Ostrowski’s classification because it induces the discrete topology and generates no interesting completion. Two absolute values are equivalent if one is a positive real power of the other (equivalently, if they define the same convergent sequences / the same topology).

3.2 The $p$-adic valuation and absolute value

Fix a prime $p$. Every nonzero rational number $x$ can be written uniquely as $$x = p^n \cdot \frac{a}{b}, \qquad n \in \mathbb{Z}, \ \ a, b \in \mathbb{Z}, \ \ p \nmid a, \ \ p \nmid b.$$ Define the $p$-adic valuation $v_p(x) := n$ (and $v_p(0) := +\infty$ by convention). This is the exact power of $p$ dividing $x$, extended sensibly to fractions — for instance $v_5(50) = 2$, $v_5(1/125) = -3$, $v_5(6/7) = 0$. The valuation satisfies: $$v_p(xy) = v_p(x) + v_p(y), \qquad v_p(x+y) \ge \min(v_p(x), v_p(y)),$$ with equality in the second whenever $v_p(x) \ne v_p(y)$ — a fact that does all the real work in this course, so hold onto it.

The $p$-adic absolute value is then $|x|_p := p^{-v_p(x)}$ (and $|0|_p := 0$). Translating the valuation properties: $|xy|_p = |x|_p |y|_p$ (multiplicative, as required) and $$|x + y|_p \le \max(|x|_p, |y|_p),$$ the ultrametric or strong triangle inequality — strictly stronger than the ordinary triangle inequality ($\max(a,b) \le a+b$ always), and the source of every strange, beautiful fact about $p$-adic geometry in Module 3.

3.3 Ostrowski’s theorem

Theorem (Ostrowski, 1916). Every nontrivial absolute value on $\mathbb{Q}$ is equivalent to $|\cdot|_\infty$ or to $|\cdot|_p$ for exactly one prime $p$.

This is the precise statement behind §1’s claim that $\mathbb{R}$ is “one place among many.” The set of places of $\mathbb{Q}$ is $\{\infty, 2, 3, 5, 7, 11, \dots\}$ — one archimedean place, one for each prime — and Ostrowski’s theorem says this list is exhaustive: there is no other way to measure distance on $\mathbb{Q}$ that respects the field operations.

3.4 The ultrametric world

The strong triangle inequality forces a cascade of facts that feel wrong the first time you see them and become obvious the second:

• Every triangle is isosceles. If $|x|_p \ne |y|_p$ then $|x+y|_p = \max(|x|_p,|y|_p)$ exactly (not just $\le$) — so among any three “side lengths” derived from three points, the two largest must be equal. There is no such thing as a scalene triangle in $p$-adic geometry.
• Every point in a ball is a center of that ball. Define the closed ball $B(a, r) = \{x : |x - a|_p \le r\}$. If $b \in B(a,r)$, then $B(b,r) = B(a,r)$ — exactly the same ball. (“If you’re inside, you’re the center” — there’s no privileged point.)
• Balls are clopen. Every ball (open or closed) is simultaneously open and closed in the $p$-adic topology. This is what makes $\mathbb{Q}_p$ totally disconnected — it has no connected subsets bigger than a point, unlike $\mathbb{R}$.
• Two balls are either disjoint or nested, never partially overlapping.

None of this is pathology to be tolerated — it’s a clean, self-consistent, different geometry, and Module 3 works through it directly.

3.5 Completion

Recall (or take on faith) how $\mathbb{R}$ is built from $\mathbb{Q}$: take Cauchy sequences of rationals with respect to $|\cdot|_\infty$, declare two sequences equivalent if their difference tends to $0$, and the equivalence classes form a new field into which $\mathbb{Q}$ embeds densely, and which is now genuinely complete (every Cauchy sequence converges). $\mathbb{Q}_p$ is built by the identical construction, verbatim, with $|\cdot|_p$ in place of $|\cdot|_\infty$: Cauchy sequences of rationals with respect to $|\cdot|_p$, modulo sequences whose difference is $p$-adically small. That’s it — that’s the entire definition. Module 2 gives two more (equivalent) concrete ways to get your hands on the resulting field, because “an equivalence class of Cauchy sequences” is a correct but unusable way to actually compute.

Where to go deeper on this section alone: Kürschák’s original generalization of completion to arbitrary valued fields is still the cleanest statement of why this construction always works, for any absolute value on any field; see the historical sources in §2 and Serre’s A Course in Arithmetic, Ch. II for a modern, terse treatment covering §3.1–3.4 in about 15 pages.

4. Coursework

Eight modules. Each has learning objectives, exposition, one fully worked numerical example, and a problem set. Full solutions are in §7 — work the problems first.

Module 1 — Absolute values and Ostrowski’s theorem

Objectives: verify $|\cdot|_p$ satisfies the absolute value axioms and the ultrametric inequality; compute $p$-adic absolute values by hand; state Ostrowski’s theorem and the product formula.

Exposition. See §3.2–3.3. One more essential fact to add to your toolkit, because it ties every place of $\mathbb{Q}$ together into a single constraint: the product formula. For any nonzero $x \in \mathbb{Q}$, $$|x|_\infty \cdot \prod_{p \text{ prime}} |x|_p = 1$$ (a finite product in practice, since $|x|_p = 1$ for all but finitely many $p$ — exactly those $p$ dividing the numerator or denominator of $x$). This says the “sizes” of $x$ at every place are not independent; they’re constrained to multiply to exactly $1$. It’s the first hint that treating all the places together (the adelic viewpoint, §2, 1930s–40s) is not just a convenience but reflects a real structural fact about $\mathbb{Q}$.

Worked example. Let $x = \frac{300}{7} = \frac{2^2 \cdot 3 \cdot 5^2}{7}$. - $v_2(x) = 2 \Rightarrow |x|_2 = 2^{-2} = 1/4$. - $v_3(x) = 1 \Rightarrow |x|_3 = 1/3$. - $v_5(x) = 2 \Rightarrow |x|_5 = 1/25$. - $v_7(x) = -1 \Rightarrow |x|_7 = 7$. - All other primes: $|x|_p = 1$. - $|x|_\infty = 300/7$.

Check the product formula: $\frac{300}{7} \cdot \frac{1}{4} \cdot \frac{1}{3} \cdot \frac{1}{25} \cdot 7 = \frac{300 \cdot 7}{7 \cdot 4 \cdot 3 \cdot 25} = \frac{2100}{2100} = 1.$ ✓.

Problem set 1. 1. Compute $|{-72}|_2$, $|{-72}|_3$, and $|{-72}|_5$. 2. Prove the ultrametric inequality $|x+y|_p \le \max(|x|_p,|y|_p)$ directly from $v_p(x+y) \ge \min(v_p(x),v_p(y))$. 3. Prove that if $|x|_p \ne |y|_p$, then $|x+y|_p = \max(|x|_p, |y|_p)$ exactly. (This is the “isosceles triangle” fact from §3.4 — prove it, don’t just cite it.) 4. Verify the product formula for $x = -\frac{18}{25}$.

Module 2 — Constructing $\mathbb{Q}_p$, three ways

Objectives: understand $\mathbb{Q}_p$ as a completion by Cauchy sequences, as an inverse limit $\lim_{\leftarrow} \mathbb{Z}/p^n\mathbb{Z}$, and via base-$p$ digit expansions — and see why the three agree.

Exposition. Three equivalent constructions, useful for different purposes:

1. Cauchy sequences (§3.5): the abstract, general-purpose construction — guarantees completeness for free, since it’s the same machine that builds $\mathbb{R}$.
2. Inverse limit. Define $\mathbb{Z}_p := \lim_{\leftarrow}_n \mathbb{Z}/p^n\mathbb{Z}$ — a compatible sequence of elements $(a_1, a_2, a_3, \dots)$ with $a_n \in \mathbb{Z}/p^n\mathbb{Z}$ and $a_{n+1} \equiv a_n \;(\mathrm{mod}\;p^n)$. This is the most computationally honest picture: a $p$-adic integer is exactly “know your value mod $p$, and mod $p^2$, and mod $p^3, \dots$, consistently, forever.” $\mathbb{Q}_p$ is then the fraction field of $\mathbb{Z}_p$ (equivalently, $\mathbb{Z}_p[1/p]$).
3. Digit expansions. Every element of $\mathbb{Z}_p$ has a unique representation $$a_0 + a_1 p + a_2 p^2 + a_3 p^3 + \cdots, \qquad a_i \in \{0, 1, \dots, p-1\},$$ converging $p$-adically (the terms shrink: $|a_i p^i|_p = p^{-i} \to 0$). Elements of $\mathbb{Q}_p$ allow finitely many negative powers of $p$ too: $\sum_{i \ge N} a_i p^i$ for some $N \in \mathbb{Z}$ (possibly negative). This is precisely the picture from §1: an ordinary decimal $\sum_{i \ge -N} a_i 10^{-i}$ extends infinitely to the right; a $p$-adic expansion extends infinitely to the left. $\mathbb{Z}$ sits inside $\mathbb{Z}_p$ as exactly the expansions that terminate (are eventually all zero, reading leftward) — well, more precisely, non-negative integers have finite expansions, and negative integers have expansions that are eventually constant at $p-1$ (see the worked example).

Worked example. Find the $5$-adic expansion of $-1$. Note $p^n - 1 = (p-1) + (p-1)p + \cdots + (p-1)p^{n-1}$ for every $n$ (just expand $(p-1)(1 + p + \cdots + p^{n-1}) = p^n - 1$). So the partial sums $S_n = 4 + 4\cdot5 + 4\cdot5^2 + \cdots + 4\cdot5^{n-1} = 5^n - 1$ satisfy $|S_n - (-1)|_5 = |5^n|_5 = 5^{-n} \to 0$. So $$-1 = 4 + 4\cdot 5 + 4\cdot 5^2 + 4 \cdot 5^3 + \cdots \quad \text{in } \mathbb{Z}_5,$$ i.e. $-1$ is the $5$-adic number “$\dots 44444$” — an infinite string of $4$’s extending left, the direct analogue of $-1 = \dots 999.0$ in a “10-adic” sense (informally; $10$ isn’t prime, but the intuition — infinitely many leftward $9$’s — is exactly the same phenomenon, and is in fact used as a popular-explainer trick for why $\dots 999 = -1$ in the $10$-adics, which do exist as a ring, just not a field, since $10$ isn’t prime).

Problem set 2. 1. Find the $7$-adic expansion of $1/2$ (hint: find $a_0 \in \{0,\dots,6\}$ with $2a_0 \equiv 1 \;(\mathrm{mod}\;7)$ first, then continue digit by digit, or use that $1/2 = (1 + 6)/2 - 3= \dots$; more directly, solve $2x \equiv 1 \;(\mathrm{mod}\;7^n)$ for increasing $n$ and read off digits). 2. Show that the integer embedding $\mathbb{Z} \hookrightarrow \mathbb{Z}_p$ is injective (i.e. no two distinct integers become $p$-adically equal) but not surjective (exhibit an explicit element of $\mathbb{Z}_p$, other than via the $-1$ example above, that cannot equal any integer). 3. Prove that every element of $\mathbb{Z}_p$ is a $p$-adic limit of a sequence of ordinary integers (i.e. $\mathbb{Z}$ is dense in $\mathbb{Z}_p$). 4. Explain concretely why the inverse-limit description $(a_1, a_2, a_3, \dots) \in \lim_{\leftarrow} \mathbb{Z}/p^n\mathbb{Z}$ and the digit-expansion description $\sum a_i p^i$ are “the same data” — what is $a_n$ in terms of the digits $a_0, \dots, a_{n-1}$?

Module 3 — The topology of $\mathbb{Q}_p$

Objectives: work directly with the ultrametric consequences from §3.4; prove $\mathbb{Z}_p$ is compact; understand total disconnectedness.

Exposition. $\mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p \le 1\}$ is the closed unit ball, and it is compact — genuinely different from $\mathbb{R}$, where closed balls (bounded closed intervals) are compact too, but the reason is different and more combinatorial here. Via the inverse-limit picture, $\mathbb{Z}_p \subset \prod_n \mathbb{Z}/p^n\mathbb{Z}$ is a closed subset of a product of finite (hence compact) discrete spaces, so it’s compact by Tychonoff’s theorem — or, more hands-on, by a digit-by-digit pigeonhole/diagonalization argument identical in spirit to Bolzano–Weierstrass. $\mathbb{Q}_p$ itself is locally compact (a union of countably many translates/scalings of the compact $\mathbb{Z}_p$) but not compact, and it is totally disconnected (§3.4: clopen balls everywhere means no nontrivial connected subsets) yet not discrete (points still have accumulation points — $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, by Problem 2.3). This combination — totally disconnected, not discrete, locally compact — is characteristic of $p$-adic-type spaces and has no analogue in $\mathbb{R}$.

Worked example. Show $B(0,1) = \mathbb{Z}_p$ and $B(1, 1/p) \subset \mathbb{Z}_p$ are either disjoint or nested with the “big” ball $\mathbb{Z}_p$, never crossing. Take $a=0, r=1$ and $b=1, s=1/p$ (radius $p^{-1}$, i.e. $|x - 1|_p \le 1/p$, i.e. $x \equiv 1 \;(\mathrm{mod}\;p)$). Every $x$ with $x \equiv 1 \;(\mathrm{mod}\;p)$ automatically has $|x|_p \le 1$ (since $x = 1 + p(\cdots)$, and both terms are in $\mathbb{Z}_p$), so $B(1,1/p) \subset B(0,1)$ — fully nested, not overlapping-but-distinct, exactly as the general clopen-balls fact in §3.4 predicts.

Problem set 3. 1. Prove directly (no citing §3.4) that if $c \in B(a,r)$ then $B(c,r) = B(a,r)$, using only the ultrametric inequality. 2. Prove $\mathbb{Z}_p$ is compact using the digit-expansion picture: given a sequence of elements of $\mathbb{Z}_p$, construct a convergent subsequence by successively passing to subsequences that agree in more and more leading digits. 3. Exhibit two closed balls in $\mathbb{Q}_p$ of the same radius that are disjoint, and two of different radii, one properly containing the other — and explain why “partially overlapping, neither nested nor disjoint” is impossible in general. 4. $\mathbb{Q}_p$ is totally disconnected, so in particular it contains no interval-like connected piece — yet it’s uncountable and every point is a limit of other points (it’s not discrete). Reconcile these two facts in your own words: what does “no small connected pieces but also no isolated points” actually look like, concretely, in the digit-expansion picture?

Module 4 — Hensel’s Lemma

Objectives: state and apply Hensel’s Lemma (both the simple-root form and the Newton-iteration form); use it to decide solvability of polynomial equations in $\mathbb{Z}_p$.

Exposition. This is the payoff for building a complete field: it lets you turn a finite check (does a polynomial have a root mod $p$?) into an exact, infinite-precision statement (does it have a root in $\mathbb{Z}_p$?).

Hensel’s Lemma (simple form). Let $f(x) \in \mathbb{Z}_p[x]$, and suppose $a \in \mathbb{Z}_p$ satisfies $f(a) \equiv 0 \;(\mathrm{mod}\;p)$ and $f'(a) \nequiv 0 \;(\mathrm{mod}\;p)$. Then there is a unique $\alpha \in \mathbb{Z}_p$ with $f(\alpha) = 0$ and $\alpha \equiv a \;(\mathrm{mod}\;p)$.

The proof is literally Newton’s method, made rigorous by the ultrametric inequality instead of real-analytic estimates: iterate $a_{n+1} = a_n - f(a_n)/f'(a_n)$, and show $|a_{n+1} - a_n|_p$ shrinks fast enough (in fact doubles the number of correct $p$-adic digits at each step, just like Newton’s method doubles decimal digits of precision over $\mathbb{R}$) that the sequence is Cauchy and its limit is an exact root. A sharper version (Hensel’s Lemma, general form) drops the “simple root” requirement $f'(a) \nequiv 0$ and instead only needs $|f(a)|_p < |f'(a)|_p^2$ — the same conclusion follows, with $\alpha$ uniquely determined by being close enough to $a$.

Worked example. Does $\sqrt{2}$ exist in $\mathbb{Z}_7$? Take $f(x) = x^2 - 2$. Mod 7: $3^2 = 9 \equiv 2$, so $a=3$ works: $f(3) = 7 \equiv 0 \;(\mathrm{mod}\;7)$. And $f'(x) = 2x$, so $f'(3) = 6 \nequiv 0 \;(\mathrm{mod}\;7)$. Hensel’s Lemma applies: there’s a unique $\alpha \in \mathbb{Z}_7$ with $\alpha \equiv 3 \;(\mathrm{mod}\;7)$ and $\alpha^2 = 2$. (There’s a second root too, $\equiv -3 \equiv 4 \;(\mathrm{mod}\;7)$, by the same argument with $a=4$ — the two square roots of $2$ in $\mathbb{Z}_7$, just as $\pm\sqrt2$ are the two square roots in $\mathbb{R}$.) One Newton step to sharpen: $a_1 = 3 - \frac{3^2-2}{2\cdot3} = 3 - \frac{7}{6}$. Working mod $49$: $6^{-1} \equiv 41 \;(\mathrm{mod}\;49)$ (since $6 \cdot 41 = 246 = 5\cdot49+1$), so $a_1 \equiv 3 - 7\cdot41 = 3 - 287 \equiv 3 - 287 + 294 = 10 \;(\mathrm{mod}\;49)$. Check: $10^2 = 100 \equiv 100-49-49 = 2 \;(\mathrm{mod}\;49)$. ✓ — so $\sqrt2 \equiv 10 \;(\mathrm{mod}\;49)$ in $\mathbb{Z}_7$, i.e. $\sqrt2 = 3 + 1\cdot7 + O(7^2)$.

Problem set 4. 1. Does $x^2 = 2$ have a solution in $\mathbb{Q}_3$? Check solvability mod $3$ first, then decide whether Hensel’s Lemma applies or definitively rules it out. 2. For which odd primes $p$ does $x^2 = -1$ have a solution in $\mathbb{Z}_p$? (Hint: this is a question about squares mod $p$ you may already know from elementary number theory — connect it explicitly to that fact.) 3. Find a cube root of $1$ in $\mathbb{Z}_7$ other than $1$ itself (i.e. a nontrivial solution of $x^3 = 1$), using Hensel’s Lemma starting from a root mod $7$. 4. Explain in your own words why Hensel’s Lemma can fail to directly decide the case $f'(a) \equiv 0 \;(\mathrm{mod}\;p)$ (the excluded case) — construct a specific small example of a polynomial and a prime where $f(a)\equiv 0$, $f'(a) \equiv 0 \;(\mathrm{mod}\;p)$, and there is genuinely no root of $f$ congruent to $a$ mod $p$ in $\mathbb{Z}_p$, to see the hypothesis is load-bearing.

Module 5 — Algebraic structure: $\mathbb{Z}_p$ as a complete DVR

Objectives: understand $\mathbb{Z}_p$ as a local ring with a unique maximal ideal and residue field $\mathbb{F}_p$; understand the structure of the unit group $\mathbb{Z}_p^\times$.

Exposition. $\mathbb{Z}_p$ is an integral domain, and $v_p$ makes it into a discrete valuation ring (DVR): every nonzero ideal is a power of the unique maximal ideal $\mathfrak{m} = p\mathbb{Z}_p = \{x : v_p(x) \ge 1\}$. It is a local ring — there is exactly one maximal ideal, so exactly one “place” left to worry about once you’re already inside it — and its residue field $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p$ is exactly the finite field you already know from modular arithmetic. Every nonzero element of $\mathbb{Q}_p$ factors uniquely as $p^n u$ with $n=v_p(x) \in \mathbb{Z}$ and $u \in \mathbb{Z}_p^\times$ (a unit, i.e. $|u|_p = 1$, i.e. $u \nequiv 0 \;(\mathrm{mod}\;p)$) — the $p$-adic analogue of scientific notation, with $p^n$ playing the role of the “exponent” and $u$ the “mantissa,” except the mantissa here is an entire $p$-adic integer, not a bounded decimal.

The unit group has a clean structure. For $p$ odd: $$\mathbb{Z}_p^\times \;\cong\; \mu_{p-1} \times (1+p\mathbb{Z}_p) \;\cong\; \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p,$$ where $\mu_{p-1}$ is the group of $(p-1)$-th roots of unity (via the Teichmüller lift, a canonical way to lift each nonzero residue mod $p$ to an exact root of unity in $\mathbb{Z}_p$, using Hensel’s Lemma on $x^{p-1}-1$), and the “$1$-units” $1+p\mathbb{Z}_p$ form a group isomorphic to $(\mathbb{Z}_p, +)$ via the $p$-adic logarithm (Module 6). For $p=2$ the statement needs a small patch ($\mathbb{Z}_2^\times \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}_2$, generated by $-1$ and the $1$-units $1+4\mathbb{Z}_2$ rather than $1+2\mathbb{Z}_2$) — the recurring pattern that $p=2$ is the annoying exception throughout this subject, exactly as it is in ordinary elementary number theory.

Worked example. Decompose $17 \in \mathbb{Z}_5^\times$ into its Teichmüller part times a $1$-unit. $17 \equiv 2 \;(\mathrm{mod}\;5)$. The Teichmüller lift $\omega(2)$ is the unique $4$th root of unity in $\mathbb{Z}_5$ congruent to $2 \bmod 5$ (since $\mu_4 = \{1,2,3,4\} \bmod 5$ under the iso $\mathbb{F}_5^\times \cong \mathbb{Z}/4$). One can compute $\omega(2) \equiv 2 + O(5)$, refined by Hensel on $x^4-1$: mod 25, need $x \equiv 2 \;(\mathrm{mod}\;5)$ with $x^4\equiv1\;(\mathrm{mod}\;25)$; try $x=7$: $7^4 = 2401 = 96\cdot25+1$, so $7^4\equiv1\;(\mathrm{mod}\;25)$ ✓, and $7\equiv2\;(\mathrm{mod}\;5)$ ✓. So $\omega(2) \equiv 7 \;(\mathrm{mod}\;25)$. Then $17/\omega(2) \equiv 17/7 \;(\mathrm{mod}\;25)$; $7^{-1}\equiv18\;(\mathrm{mod}\;25)$ (since $7\cdot18=126=5\cdot25+1$), so $17\cdot18=306\equiv306-12\cdot25=6\;(\mathrm{mod}\;25)$ — and $6 \equiv 1 \;(\mathrm{mod}\;5)$ as required of a $1$-unit. So $17 = \omega(2) \cdot u$ with $\omega(2) \equiv 7$, $u \equiv 6 \;(\mathrm{mod}\;25)$, both refinable to arbitrary precision.

Problem set 5. 1. Prove $\mathbb{Z}_p$ is a local ring directly: show every element not in $p\mathbb{Z}_p$ is a unit (has a multiplicative inverse in $\mathbb{Z}_p$). 2. List all the roots of unity in $\mathbb{Q}_7$. (Use the unit group structure theorem — how many roots of unity of each possible order can there be, and why does $7-1=6$ matter here?) 3. Explain concretely why $p=2$ breaks the clean $\mu_{p-1}\times(1+p\mathbb{Z}_p)$ splitting — what goes wrong if you try to apply Hensel’s Lemma to $x^{p-1}-1 = x^0 - 1$ style reasoning when $p=2$? 4. True or false, with justification: every element of $\mathbb{Z}_p^\times$ is a root of unity times a power of $p$ times… — actually, is every unit itself expressible as (root of unity) $\times$ (element $1 + p\mathbb{Z}_p$), uniquely? What breaks if you drop uniqueness?

Module 6 — $p$-adic analysis: series, exp, and log

Objectives: state and use the (surprisingly simple) $p$-adic series convergence test; define $\exp_p$ and $\log_p$ and know their radii of convergence; contrast sharply with real analysis.

Exposition. Here is the single most pleasant fact in elementary $p$-adic analysis, a direct payoff of the ultrametric inequality: a series $\sum_{n=0}^\infty a_n$ of $p$-adic numbers converges if and only if $a_n \to 0$. (Contrast with $\mathbb{R}$: $a_n \to 0$ is necessary but famously not sufficient — the harmonic series $\sum 1/n \to 0$ termwise but diverges. No such counterexample can exist $p$-adically; the ultrametric inequality applied to partial sums makes “terms shrink to zero” automatically force the partial sums to be Cauchy.) This single fact makes checking convergence of $p$-adic series dramatically more mechanical than in ordinary analysis — you never need ratio tests, comparison tests, or alternating series tricks; you just check $v_p(a_n) \to \infty$.

The $p$-adic exponential and logarithm are defined by the same power series as their real/complex counterparts, but their convergence behavior is quite different because of the factorials involved (and $v_p(n!)$ grows, but only like $n/(p-1)$, by Legendre’s formula, not fast enough to guarantee convergence everywhere the way it does over $\mathbb{C}$): $$\exp_p(x) = \sum_{n=0}^\infty \frac{x^n}{n!} \quad \text{converges for } v_p(x) > \tfrac{1}{p-1},$$ $$\log_p(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n} \quad \text{converges for } v_p(x) \ge 1.$$ For $p$ odd, $\frac{1}{p-1} \le \frac12 < 1$, so $\exp_p$ converges (at least) on all of $p\mathbb{Z}_p$; for $p=2$ you need the sharper $v_2(x) \ge 2$, i.e. $x \in 4\mathbb{Z}_2$ — the $p=2$ exception again. On their common domain of convergence, $\exp_p$ and $\log_p$ are mutually inverse and give the isomorphism $(1+p\mathbb{Z}_p, \times) \cong (p\mathbb{Z}_p, +) \cong (\mathbb{Z}_p,+)$ promised in Module 5.

Worked example. Does $_{n=1}^n! $ converge in $\mathbb{Q}_p$ for any $p$? No series test needed beyond the one fact above: $v_p(n!) \to \infty$ as $n \to \infty$ (Legendre’s formula: $v_p(n!) = \sum_{i\ge1} \lfloor n/p^i \rfloor \ge (n - s_p(n))/(p-1) \to \infty$), so $|n!|_p \to 0$, so yes, $\sum n!$ converges $p$-adically for every prime $p$ — a series that diverges (to $\infty$) in every conceivable real sense converges perfectly well $p$-adically. This is a genuinely good example to keep in your back pocket for how differently “convergence” behaves in the two worlds.

Problem set 6. 1. Determine, for each prime $p$, whether $\sum_{n=1}^\infty p^n/n$ converges in $\mathbb{Q}_p$. 2. Compute $\log_p(1+p)$ to three terms (as a series in powers of $p$) for $p=5$, and simplify as far as you can. 3. Prove the “term test” (convergence $\iff a_n \to 0$) for $p$-adic series, using only the ultrametric inequality applied to $S_m - S_n$ for $m > n$. 4. Contrast, explicitly, why $\sum 1/n$ diverges in $\mathbb{R}$ despite $1/n \to 0$, but the analogous danger cannot occur in $\mathbb{Q}_p$ — what property of the ultrametric inequality is doing the work that the ordinary triangle inequality can’t provide?

Module 7 — The local-global principle

Objectives: state the Hasse–Minkowski theorem; work a concrete local-solvability check; understand that the principle is special to quadratic forms via the Selmer counterexample.

Exposition. Theorem (Hasse–Minkowski). A quadratic form $Q(x_1,\dots,x_n)$ with rational coefficients has a nontrivial zero in $\mathbb{Q}^n$ if and only if it has a nontrivial zero in $\mathbb{R}^n$ and a nontrivial zero in $\mathbb{Q}_p^n$ for every prime $p$.

The remarkable content: checking infinitely many local conditions is, in practice, a finite computation — solvability mod $p$ for all but finitely many “bad” primes (those dividing the discriminant of the form) is automatic by Hensel’s Lemma applied to the smooth locus, so you only need to check $\mathbb{R}$ and the finitely many bad primes directly. This “local conditions determine global existence” phenomenon is the philosophical ancestor of an enormous amount of modern arithmetic geometry.

It is emphatically not universal, though, and knowing where it breaks is as important as knowing where it holds. Selmer’s example (1951): the cubic curve $$3x^3 + 4y^3 + 5z^3 = 0$$ has a nontrivial solution over $\mathbb{R}$ and over $\mathbb{Q}_p$ for every prime $p$ — yet has no nontrivial rational solution. Local-global fails outright once you leave quadratic forms. This failure is exactly what the Selmer group in elliptic curve descent theory (mentioned but not built out in your elliptic curves course, Module 5’s descent sketch) is built to measure — the Selmer group is precisely the obstruction group that’s nonzero exactly when local solvability everywhere fails to imply global solvability, for the specific descent equations that arise from halving points on an elliptic curve. Selmer’s own counterexample and the group named after him are the same underlying phenomenon, applied to two different-looking problems thirty years apart.

Worked example. Is $5x^2 + 7y^2 = 13z^2$ nontrivially solvable over $\mathbb{Q}$? Check $\mathbb{R}$: yes trivially (indefinite form, e.g. plenty of real solutions by continuity/sign considerations — take $z$ large). Check small primes dividing the coefficients ($5,7,13$) and $2$: mod $5$, need $7y^2 \equiv 13z^2$, i.e. $2y^2\equiv3z^2 \;(\mathrm{mod}\;5)$; try $z=1$: need $2y^2\equiv3$, i.e. $y^2\equiv3\cdot3=9\equiv4$ (since $2^{-1}\equiv3\bmod5$), so $y\equiv\pm2$ works — solvable mod 5, and since the relevant partial derivative is nonzero here, Hensel lifts this to a genuine solution in $\mathbb{Q}_5$. Similarly one checks $\mathbb{Q}_7$ and $\mathbb{Q}_{13}$ (both solvable by the same style of check) and $\mathbb{Q}_2$. Since all local conditions hold, Hasse–Minkowski guarantees a rational solution exists — and indeed one can find it directly: $(x,y,z)=(6,1,5)$ gives $5\cdot36+7\cdot1=180+7=187$ and $13\cdot25=325$ — that particular guess doesn’t work, which is the point of the exercise (verifying local solvability is a genuine computation, not a formality) — Problem 1 below asks you to finish the job properly.

Problem set 7. 1. Finish checking local solvability of $5x^2+7y^2=13z^2$ at all remaining relevant primes, and (harder, optional) search for an explicit rational solution guaranteed to exist by Hasse–Minkowski. 2. Explain precisely why checking “all but finitely many primes” is automatic for a quadratic form — which theorem from Module 4 is doing that work, and what property of the form at a “good” prime (one not dividing the discriminant) guarantees its hypotheses hold? 3. Selmer’s cubic has no rational point despite passing every local test. In your own words, explain what this tells you about the logical status of the local-global principle — is it a general theorem about $\mathbb{Q}$, or a special feature of quadratic forms specifically? What class of equations would you now be suspicious of, going in? 4. Conceptual, linking back to the elliptic curves course: the Selmer group of an elliptic curve $E/\mathbb{Q}$ sits in an exact sequence $0 \to E(\mathbb{Q})/2E(\mathbb{Q}) \to \mathrm{Sel}_2(E) \to Ш(E)[2] \to 0$, where $Ш$ (the Tate–Shafarevich group) is, essentially by definition, the group measuring exactly the Selmer-cubic-style failures for the specific curve $E$. Given everything above, explain in your own words why computing the rank of $E(\mathbb{Q})$ (Module 5 of the elliptic curves course) is fundamentally a local-global question, and why $Ш$ being potentially nonzero is what makes rank computation hard in general.

Module 8 — Modern $p$-adic number theory: Iwasawa theory and Galois representations

Objectives: understand, at a sketch level, what a $p$-adic $L$-function is and why it’s natural; understand why $\mathbb{Z}_p$ is the right coefficient ring for the Galois representations behind Wiles’s proof.

Exposition. Classical fact (Kummer, 19th century, well before $p$-adic numbers existed as objects): certain congruences hold between values of the Riemann zeta function at negative integers (equivalently, between Bernoulli numbers) modulo a prime $p$, whenever the indices are congruent mod $p-1$. These Kummer congruences look, in hindsight, exactly like the statement “a certain function is $p$-adically continuous, sampled only at integer points” — and that is precisely what Kubota and Leopoldt made precise in 1964, constructing an honest $p$-adic $L$-function $L_p(s,\chi)$, a genuine continuous (in fact $p$-adic analytic) function on a $p$-adic domain, that interpolates the classical values $\zeta(1-n)$ at negative integers, packaging infinitely many classical congruences into one analytic object living natively in $\mathbb{Q}_p$. This is the general pattern of Iwasawa theory: take a sequence of classical arithmetic invariants (class numbers, $L$-values, ranks of elliptic curves) indexed by integers or by fields in a tower, and show they’re literally the values of one $p$-adic analytic function, which then lets you use $p$-adic analysis (Module 6 — much simpler convergence behavior!) to prove things about the whole infinite family at once instead of one case at a time.

The other modern pillar, and the one that connects hardest back to your elliptic curves course: a Galois representation attached to an elliptic curve $E/\mathbb{Q}$ (or a modular form) is a continuous homomorphism $\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(R)$ for a suitable coefficient ring $R$. The mod $p$ version has $R = \mathbb{F}_p$, and is comparatively simple (finite target, easy to compute with). Wiles’s strategy for proving modularity was to start from a mod-$p$ representation known to be modular, and deform it — that is, study the entire space of $p$-adic ($R=\mathbb{Z}_p$, or more generally $R$ a complete local $\mathbb{Z}_p$-algebra) representations reducing to that fixed mod-$p$ one — and show, via the celebrated “$R = \mathbb{T}$” theorem (comparing a deformation ring to a Hecke algebra), that every such deformation is modular. $\mathbb{Z}_p$ isn’t a minor technical convenience in this story — it is the only reason the deformation space (an honest, structured, $p$-adically-complete geometric object) exists at all for the argument to act on. Without the $p$-adic numbers built in Modules 2–3 of this course, there is no space to deform inside.

Worked example. A hint of the Kummer congruence phenomenon itself, using small actual Bernoulli numbers ($B_2=\frac16, B_4=-\frac1{30}, B_6=\frac1{42}, B_8=-\frac1{30}, B_{10}=\frac{5}{66}, B_{12}=-\frac{691}{2730}$): take $p=5$. Kummer’s congruence predicts $\frac{B_n}{n} \equiv \frac{B_m}{m} \;(\mathrm{mod}\;5)$ whenever $n \equiv m \;(\mathrm{mod}\;4)$ (since $p-1=4$) and $4 \nmid n$. Check $n=6, m=2$ (both $\equiv2 \bmod4$): $B_6/6 = \frac{1}{252}$ and $B_2/2 = \frac{1}{12}$. Working $5$-adically ($252 = 4\cdot63=4\cdot9\cdot7$, coprime to 5; $12$ coprime to 5, so both are $5$-adic units): $\frac1{252} - \frac1{12} = \frac{12-252}{252\cdot12} = \frac{-240}{3024}$, and $-240 = -5\cdot48$, so $v_5(-240)=1>0$ while the denominator is a unit — so indeed $\frac{B_6}{6}\equiv\frac{B_2}{2}\;(\mathrm{mod}\;5)$. ✓ — a finite, checkable instance of the exact congruence phenomenon that motivated the entire construction of $p$-adic $L$-functions.

Problem set 8. 1. Check the Kummer congruence prediction for $n=10, m=6$ at $p=5$ (again $10\equiv6\equiv2\bmod4$) using the Bernoulli numbers given above. 2. In your own words, explain the difference between what a classical $L$-function encodes (an analytic function of a complex variable $s$) and what a $p$-adic $L$-function encodes (a $p$-adic analytic function of a $p$-adic variable) — what exactly is being “reorganized” when you pass from one to the other, and what do they have in common (what values do they share)? 3. Why does it make sense, given Module 6’s exp/log story ($1+p\mathbb{Z}_p \cong \mathbb{Z}_p$ as topological groups), that a “$p$-adic variable” can meaningfully vary continuously the way a complex variable does — what plays the role of $\mathbb{C}$’s connectedness/openness here, given that $\mathbb{Q}_p$ is totally disconnected (Module 3)? 4. Research/synthesis question: look up the current status of the Iwasawa Main Conjecture (proved by Mazur–Wiles in 1984 for $\mathbb{Q}$, with various generalizations since) and its relationship to the $p$-adic Birch and Swinnerton-Dyer conjecture. How does this compare, in terms of “proved vs. open,” to the status of the classical BSD conjecture from your elliptic curves course, Module 5?

5. Resources, organized

Textbooks — roughly easy → hard: - Fernando Q. Gouvêa, p-adic Numbers: An Introduction — the best true starting point, covers almost exactly §1–§6 of this course at a leisurely pace with excellent motivation. Publisher page. - Jean-Pierre Serre, A Course in Arithmetic — Chapter II builds $p$-adic fields in about 15 dense, beautiful pages; the rest of the book (quadratic forms, then modular forms) is itself a masterclass in the local-global philosophy of Module 7. - Neal Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions — the natural next step after Gouvêa, leaning toward Module 6 and Module 8 material (includes a construction of $p$-adic zeta functions). - Neal Koblitz, p-adic Analysis: A Short Course on Recent Work (LMS Lecture Notes) — more advanced, aimed directly at the Iwasawa-theory/$p$-adic-$L$-function material of Module 8. Publisher listing. - J.W.S. Cassels, Local Fields — the classical rigorous graduate reference for everything in Modules 1–5 and local class field theory beyond this course’s scope.

Free lecture notes / courses: - Andrew Sutherland (and, in other years, Bjorn Poonen), MIT 18.785 Number Theory I — outstanding free lecture notes that build local fields and $p$-adic numbers rigorously as part of a full graduate number theory course; look for the lectures on absolute values, completions, and local fields specifically. Full lecture notes, Fall 2021, most recent iteration’s lecture index. - Keith Conrad’s expository notes — short, exceptionally clear, and aimed at exactly the sticking points a first-time learner hits. See especially “The $p$-adic Expansion of Rational Numbers” and “Infinite Series in $p$-adic Fields.” Full index at kconrad.math.uconn.edu/blurbs, The p-adic expansion of rational numbers, Infinite series in p-adic fields. - Chris Williams, An Introduction to p-adic L-functions (parts I & II) — free, modern, self-contained lecture notes that go from scratch to Kubota–Leopoldt and beyond, directly extending Module 8. Part I, Part II: modular forms. - Joaquín Rodrigues Jacinto & Chris Williams, “An introduction to $p$-adic $L$-functions,” a 2023 expository survey (arXiv) — a more recent, very readable overview of the whole area. arXiv:2309.15692. - Brian Conrad, CMI Summer School Notes on p-adic Hodge Theory — the standard free entry point into the Module 8 frontier material (Fontaine’s rings, comparison theorems). Notes PDF. - numbertheory.org’s lecture notes index — a broad, curated list of free number theory course notes across many schools and levels. numbertheory.org/ntw/lecture_notes.html.

Historical / primary sources (for §2): - Fernando Q. Gouvêa, “Hensel’s $p$-adic Numbers: Early History” — a historian-mathematician’s account of exactly how and why Hensel built these numbers, written for a modern audience. PDF. - MacTutor History of Mathematics, “Kurt Hensel” biography. mathshistory.st-andrews.ac.uk/Biographies/Hensel. - Wikipedia’s “P-adic number”, “P-adic valuation”, and “P-adic analysis” pages are all well-sourced starting points into the primary literature.

Interactive / computational tools: - LMFDB — $p$-adic (local number) fields database — look up any local field, its ramification data, Galois group, and more; the direct local analogue of the elliptic curve database you already used in the other course. Individual example: $p$-adic field 3.6.7.3. - SageMath — Introduction to the $p$-adics — free, runs in-browser via SageCell, lets you construct $\mathbb{Q}_p$, compute Hensel lifts, and check every hand computation in Modules 1–6 in seconds.

6. Connection graph — where this fits in your map

Same purpose as §6 of the elliptic curves course: forward pointers, not a closed diagram. Note the explicit join back to elliptic-curves-course.md at three separate nodes — this is not a coincidence; the two courses are genuinely one story split into two documents for manageability.

How to read this as your next moves, in rough order of directness:

• Algebraic number theory more broadly (rings of integers, ramification, class groups, global class field theory) is the natural companion thread if §2–§3 (valuations, Ostrowski, local fields) was the most satisfying part — it’s the global counterpart to everything local built here, and the two halves (local + global, glued by adeles) are how professional number theorists actually think.
• Iwasawa theory and $p$-adic $L$-functions, then Galois representations and the Langlands program, is the natural next thread if Module 8 was the draw — this is the most direct route back into the Wiles storyline from your elliptic curves course, but now from the $p$-adic-analytic side rather than the modular-forms side.
• $p$-adic Hodge theory and rigid/Berkovich analytic geometry is the thread if Module 6 (p-adic analysis) and the ultrametric topology of Module 3 were the draw — this is genuinely a full alternate universe of “geometry,” parallel to complex analytic geometry, and one of the most active current research areas (Scholze’s perfectoid spaces).
• The local-global principle and Diophantine geometry (Chabauty’s method, which uses $p$-adic analysis directly to bound rational points on curves — another explicit bridge to your elliptic curves course) is the applied-to-classical-problems thread if Module 7 was the draw.
• Model theory of valued fields (the Ax–Kochen theorem, which uses $p$-adic fields to resolve a question originally about $\mathbb{Z}$, via a completely different toolkit — mathematical logic) is a left-field but genuinely rewarding thread if the “totally disconnected, non-discrete” topology of Module 3 fascinated you and you’re curious what other fields find that structure useful for.

As with the elliptic curves course: when you pick a next node, start the new document by naming which node you’re standing on and what you already know from here — that’s the whole mechanism this project is built around.

7. Answer key

Work the problems first. Answers are terse — they confirm the destination, not a substitute for the derivation.

Module 1 1. $-72 = -1 \cdot 2^3 \cdot 3^2$. $|{-72}|_2 = 2^{-3} = 1/8$; $|{-72}|_3 = 3^{-2}=1/9$; $|{-72}|_5 = 5^0=1$ (5 doesn’t divide 72). 2. $v_p(x+y)\ge\min(v_p(x),v_p(y)) \Rightarrow p^{-v_p(x+y)} \le p^{-\min(v_p(x),v_p(y))} = \max(p^{-v_p(x)},p^{-v_p(y)})$, i.e. $|x+y|_p \le \max(|x|_p,|y|_p)$. 3. WLOG $v_p(x) < v_p(y)$. Write $x+y$; suppose for contradiction $v_p(x+y) > v_p(x)$. Then $x = (x+y) - y$ has $v_p(x) \ge \min(v_p(x+y), v_p(y)) > v_p(x)$ (since both terms exceed $v_p(x)$ by assumption/given), a contradiction. So $v_p(x+y) = v_p(x) = \min$, giving $|x+y|_p = |x|_p = \max(|x|_p,|y|_p)$. 4. $-18/25 = -1\cdot2\cdot3^2\cdot5^{-2}$. $|{\cdot}|_2=1/2$, $|{\cdot}|_3=1/9$, $|{\cdot}|_5=25$, all other primes $1$, $|{\cdot}|_\infty = 18/25$. Product: $\frac{18}{25}\cdot\frac12\cdot\frac19\cdot25 = \frac{18\cdot25}{25\cdot2\cdot9} = \frac{18}{18}=1$ ✓.

Module 2 1. Solve $2x\equiv1\;(\mathrm{mod}\;7^n)$: mod 7, $x=4$ ($2\cdot4=8\equiv1$). Mod 49: $2x\equiv1\;(\mathrm{mod}\;49)$, $x=25$ ($2\cdot25=50\equiv1$), and $25 \equiv 4\;(\mathrm{mod}\;7)$ ✓ consistent. Continuing gives $1/2 = 4 + 3\cdot7 + 3\cdot7^2+\cdots$ (digit pattern stabilizes at $3$ after the first digit, since $1/2 = (1+6)/2 = $ … equivalently $1/2 = -3 = -3$, and one can directly find $-3$’s expansion: $-3 = 4 + 3\cdot7+3\cdot7^2+\cdots$ by the same telescoping trick as the $-1$ example, since $1/2 \equiv -3 \;(\mathrm{mod}\;7^n)$ for all $n$ isn’t quite right — the clean way: $1/2 = -3$ is false in $\mathbb{Q}$, but $2\cdot(-3)=-6\equiv1\;(\mathrm{mod}\;7)$; check $2\cdot(-3)=-6$, and $-6\equiv1\;(\mathrm{mod}\;7)$ ✓, so digit-by-digit $1/2$ and $-3$ agree — giving $1/2 = 4+3\cdot7+3\cdot7^2+3\cdot7^3+\cdots$ in $\mathbb{Z}_7$). 2. Injective: if integers $m\ne n$ agree $p$-adically they’d have $v_p(m-n)=\infty$, forcing $m=n$ in $\mathbb{Z}$, contradiction. Not surjective: $-1 = \sum(p-1)p^i$ from the worked example is not equal to any single integer (its expansion never terminates and never eventually-stabilizes at $0$, which is required of nonneg integers, nor does it equal a negative integer other than via the specific eventually-constant-at-$(p-1)$ pattern matching that integer — more simply, any element whose digit sequence is not eventually periodic-with-period-dividing-appropriately / not eventually constant at $0$ or forced pattern is a non-integer, e.g. $\sum_{i\ge0} p^{2^i}$-style sparse sums, or simply any square root from Module 4 like $\sqrt2 \in \mathbb{Z}_7$, which is irrational hence not an integer). 3. Given $x=\sum_{i\ge0}a_ip^i \in \mathbb{Z}_p$, the truncations $x_n = \sum_{i=0}^{n-1}a_ip^i \in \mathbb{Z}$ satisfy $|x-x_n|_p \le p^{-n}\to0$, so $x_n\to x$: an explicit sequence of integers converging to $x$. 4. $a_n \in \mathbb{Z}/p^n\mathbb{Z}$ is exactly the truncated sum $a_0+a_1p+\cdots+a_{n-1}p^{n-1} \bmod p^n$ — the compatibility condition $a_{n+1}\equiv a_n \;(\mathrm{mod}\;p^n)$ is exactly the statement that extending the digit expansion by one more digit doesn’t change any of the earlier digits, which is obviously true and is exactly what “reading off more terms of a convergent series” means.

Module 3 1. For any $x \in B(c,r)$: $|x-a|_p = |(x-c)+(c-a)|_p \le \max(|x-c|_p,|c-a|_p) \le \max(r,r)=r$ (using $c\in B(a,r)$ so $|c-a|_p\le r$), so $x\in B(a,r)$; the reverse inclusion is symmetric. Hence $B(c,r)=B(a,r)$. 2. Given a sequence $(x^{(k)})$ in $\mathbb{Z}_p$, infinitely many share the same digit $a_0$ (pigeonhole, only $p$ choices); pass to that subsequence; among those, infinitely many share $a_1$ too; continue. Diagonalize: pick one element from each nested subsequence to build a genuine subsequence of the original whose digits agree with a fixed limiting digit string to increasing precision — that subsequence converges (by construction) to the $p$-adic number with those digits. 3. Same radius, disjoint: $B(0,1/p)$ and $B(1,1/p)$ in $\mathbb{Z}_p$ (residues $0$ and $1$ mod $p$ are different, and balls of radius $1/p$ are exactly residue classes mod $p$ — disjoint since $0\ne1 \bmod p$). Different radii, nested: $B(1,1/p^2) \subset B(1,1/p)$ (a finer residue class mod $p^2$ sits inside the coarser one mod $p$ that it refines). Partial overlap is impossible because if two balls shared even one point, Problem 1 shows that point is a valid center for both — and then the smaller ball’s every point is within the smaller radius of that shared center, hence within the larger radius too, forcing full containment. 4. Concretely: fixing finitely many leading digits carves out a clopen ball, and every element in that ball differs from every other only in digits far enough out — so any two distinct elements are eventually separated at some finite digit, meaning there’s always a clopen set containing one but not the other arbitrarily close by (no interval-like continuum to be connected), yet you can also always find another element agreeing in arbitrarily many leading digits (perturb only a far-out digit), so no point is isolated either. “Totally disconnected but not discrete” is exactly “differences show up at some finite but unbounded digit position.”

Module 4 1. Mod 3: squares are $\{0,1\}$; $2$ is not a square mod 3, and since $2$ is a unit ($3\nmid2$), Hensel’s non-solvability is definitive here (unit square-ness mod $p$ for odd $p$ exactly determines square-ness in $\mathbb{Z}_p$) — so no, $x^2=2$ has no solution in $\mathbb{Q}_3$. 2. $x^2=-1$ solvable in $\mathbb{Z}_p$ (p odd) iff $-1$ is a square mod $p$ (Hensel applies directly since $f'(x)=2x\ne0$ whenever $x\ne0$, and $x=0$ can’t be a root since $-1\ne0$) iff $p\equiv1\;(\mathrm{mod}\;4)$ — exactly the classical elementary number theory fact about which primes are sums of two squares / have $-1$ as a QR. 3. Mod 7: need $a^3\equiv1$, $a\ne1$. Try $a=2$: $8\equiv1\;(\mathrm{mod}\;7)$ ✓. $f(x)=x^3-1$, $f'(x)=3x^2$, $f'(2)=12\equiv5\nequiv0\;(\mathrm{mod}\;7)$, so Hensel applies: a genuine cube root of unity $\alpha\equiv2\;(\mathrm{mod}\;7)$ exists in $\mathbb{Z}_7$ (and $\alpha=4$ gives the third one, since $1,2,4$ are exactly the cube roots of unity mod 7, matching $\mu_3 \subset \mathbb{F}_7^\times$ as $3\mid6=7-1$). 4. Take $p=2$, $f(x)=x^2-1$ near $a=1$: $f(1)=0\equiv0\;(\mathrm{mod}\;2)$ but $f'(1)=2\equiv0\;(\mathrm{mod}\;2)$ too — hypothesis fails. Here it happens $x=1$ genuinely is a root, so this particular case doesn’t demonstrate failure; a cleaner failing example: $f(x)=x^2+1$ over $p=2$ at $a$ where $f(a)\equiv0\;(\mathrm{mod}\;2)$ (need $a$ odd, $a=1$: $f(1)=2\equiv0\;(\mathrm{mod}\;2)$) but $f'(1)=2\equiv0\;(\mathrm{mod}\;2)$ — and indeed $x^2=-1$ has no solution in $\mathbb{Z}_2$ (check mod 4: odd squares are $\equiv1\;(\mathrm{mod}\;4)$, never $\equiv-1\equiv3\;(\mathrm{mod}\;4)$) — confirming the simple form of Hensel’s Lemma genuinely cannot be dropped.