Berkeley Math Circle. September 24, 2000.

A. Givental. Introduction to $p$-adic numbers

Definition. A norm on the field ${\Bbb Q}$ of rational numbers is a function $r\mapsto \Vert r \Vert$ taking values in the set of non-negative real numbers satisfying (i) $\Vert r \Vert >0$ unless $r=0$; (ii) $\Vert rr'\Vert =\Vert r\Vert \cdot \Vert r'\Vert$; (iii) $\Vert r+r'\Vert \leq \Vert r\Vert + \Vert r'\Vert$.

Examples of norms: (a) $\Vert r \Vert _{\infty}:= \vert r\vert$. (b) For any prime $p$ put $\Vert r\Vert _p =1/p^k$ if $r=p^k m/n$ where $k$ is an integer, and $m,n$ are not divisible by $p$. (c) The trivial norm $\Vert r\Vert=1$ unless $r=0$. (d) Norms equivalent to $\Vert \cdot \Vert _p$ are obtained as $\Vert \cdot \Vert _p^{\alpha}$ where $\alpha >0$ for $p\neq \infty$ and $1>\alpha >0$ for $p=\infty$.

Theorem (Ostrovsky). Any non-trivial norm on ${\Bbb Q}$ is equivalent to one of $\Vert \cdot \Vert _p$ with $p=2,3,5,7,...,\infty$.

Given a norm, one measures distances between $x$ and $y$ as $\Vert x-y \Vert$. The field ${\Bbb Q}_p$ of $p$-adic numbers is constructed as the completion of ${\Bbb Q}$ with respect to the norm $\Vert \cdot \Vert _p$. Informally speaking, it consists of all ``things'' which can be approximated by rationals as $\Vert \cdot \Vert _p$-closely as desired. In fact we have ${\Bbb Q}_{\infty}={\Bbb R}$, the field of real numbers.

For $p\neq \infty$, consider sequences $a = ... a_n ... a_2 a_1 a_0 . a_{-1} ... a_{-m}$, of base-$p$ digits, $a_i =0,1,...,p-1$, infinite to the left. One can perform addition, subtraction, multiplication and division following the middle-school rules for base-$p$ numbers. With these operations, the set of all such sequences forms the field of $p$-adic numbers ${\Bbb Q}_p$. One extends the norm to $p$-adic numbers by $\Vert a\Vert _p:= p^{-k}$ where $k$ is the position of the rightmost non-zero digit $a_k$.

Problems.

1. (a) Show that any norm satisfies $\Vert 0\Vert=0, \Vert\pm 1\Vert=1$.

(b) Explain in words the meaning of the inequality $\Vert r \Vert _p \leq 1$ for a rational number $r$.

(c) Show that a rational number $r$ satisfying $\Vert r \Vert _p \leq 1$ for all primes $p$ is an integer.

(d) Prove that $\Vert p^N!\Vert _p=(p^N-1)/(p-1).$ Let $s_p(n)$ denote the sum of base-$p$ digits of $n$. Check that $n-s_p(n)$ is divisible by $p-1$. Put $k_p(n)=(n-s_p(n))/(p-1)$ and prove that $\Vert n! \Vert _p=1/p^{k_p(n)}$.

2. Show that in ${\Bbb Q}_p$

(a) $p^n\to 0$ when $n\to \infty$.

(b) For any integers $a_k$ the series $a_0+a_1p+a_2p^2+...+a_kp^k+...$ converges.

(c) $-1 = .... a a a a a . 0$ where $a=p-1$.

(d) Rational numbers are characterized as sequences of base-$p$ digits infinite to the left and periodic beginning with some place.

(e) The series $\ln (1+x) = x-x^2/2+x^3/3-x^4/4+...$ and $\exp (x) = 1+x+x^/2+...+x^n/n!+...$ converge if and only if $\Vert x\Vert _p<1$.

3. (a) Show that in ${\Bbb Q}_5$ the equation $x^2=a$ has a solution if and only if the rightmost non-zero digit $a_k$ of $a$ has even position $k$ and is equal to $1$ or $4$.

(b) Compute a few last digits of $\sqrt{-1}$ in ${\Bbb Q}_5$. How many such roots are there?

(c) Describe $p$-adic numbers which have square roots in ${\Bbb Q}_p$ for $p\neq 2$; for $p=2$.

(d) For which $5$-adic numbers $a,b$ does the equation $x^2/a+y^2/b=1$ has a solution $(x,y)$ in ${\Bbb Q}_5$?

4. Denote ${\Bbb Z}_p$ the set of $p$-adic integers, that is $p$-adic numbers $a$ with $\Vert a\Vert _p \leq 1$.

(a) Check that all rational integer numbers are $p$-adic integers.

(b) Which rational numbers are $p$-adic integers?

(c) Show that $p$-adic integers can be approximated by rational integers as $\Vert \cdot \Vert _p$-precisely as desired.

(d) Prove that any infinite sequence of $p$-adic integers has a subsequence convergent in ${\Bbb Z}_p$.

(e) Let $F(x)$ be a polynomial with integer rational coefficients. Derive from (d) that the equation $F(x)=0$ has a solution in ${\Bbb Z}_p$ if and only if the congruence $F(x)\equiv 0 (modulo p^k)$ has a solution for each $k=1,2,3,...$.

5. Let $a$ be an integer rational number. Prove that the sequence $a^{p^n}$ converges in ${\Bbb Z}_p$ when $n\to \infty$, and that the limit, denoted $sgn_p(a)$ and called $p$-adic sign of $a$, satisfies $sgn_p(a)\equiv a (modulo p)$, $sgn_p(a)^{p}=sgn_p(a)$ and $sgn_p(ab)=sgn_p(a) sgn_p(b)$. Derive that the equation $x^p=x$ has $p$ distinct solutions in ${\Bbb Z}_p$. Examine the example $p=5$.

6. (a) Introduce ${\Bbb Z}_{10}$ using decimal digits and their sequences $... a_2 a_1 a_1 .$ infinite to the left. Identify ${\Bbb Z}_{10}$ with the Cartesian product ${\Bbb Z}_2\times {\Bbb Z}_5$ equipped with component-wise operations of addition and multiplication.

(b) Show that for any natural $k$ there exist exactly four $k$-digit endings,
$...000000.\ \ ...000001.\ \ ...890625.\ \ ...109376. \$, which reproduce themselves under multiplication (that is $N_1N_2$ has this ending whenever both $N_1$ and $N_2$ do).

7. The Hilbert symbol $(a,b)_p$ is defined to be $1$ if the equation $x^2/a+y^2/b=1$ has a solution $(x,y)$ in ${\Bbb Q}_p$ and $-1$ if it does not have a solution in ${\Bbb Q}_p$.

(a) Show that $(a,b)_p=1$ if at least one of $a,b$ has square roots in ${\Bbb Q}_p$, that $(a,b)_p=(b,a)_p$, $(a,-a)_p=1$, and $(a,b)_{\infty}=-1$ if and only if both $a,b<0$.

(b) Prove that $(a,bc)_p=(a,b)_p(a,c)_p$.

(c) Show that (a) and (b) reduce computation of Hilbert symbols to that of $(p,a)_p$ and $(b,c)_p$ with $\Vert a \Vert _p=\Vert b \Vert _p=\Vert c \Vert _p=1$.

(d) Show that for $p\neq 2$ and $a,b,c$ as in (c) we have $(p,a)_p=-1$ if and only if $a$ has no square roots in ${\Bbb Q}_p$, and $(b,c)_p=1$.

(e) Show that for given $a,b$ the set of primes $p$ such that $(a,b)_p=-1$ is finite.

Theorem (Hilbert). $(a,b)_2(a,b)_3 ... (a,b)_p ... (a,b)_{\infty} =1$.

Theorem (Hasse - Minkovsky). The equation $x^2/a+y^2/b=1$ has a solution $(x,y)$ in ${\Bbb Q}$ if and only if it has a solution in ${\Bbb Q}_p$ for each $p=2,3,...,\infty$ (in other words -- if $(a,b)_p=1$ for all $p=2,3,...,\infty$).