Chapter 12

Let \(M_{0}\) be the set of absolute values consisting of the ordinary absolute value and all p-adic absolute values \(v_{p}\) on the field of rational numbers Q. Show that for any rational number \(a \in \mathbf{Q}, a \neq 0\), we have $$ \prod_{\text {e? } \mathrm{M}_{\mathbf{Q}}}|a|_{v}=1 $$ If \(K\) is a finite extension of \(\mathbf{Q}\), and \(M_{K}\) denotes the set of absolute values on \(K\) extending those of \(M_{\mathbf{Q}}\), and for each \(w \in M_{K}\) we let \(N_{w}\) be the local degree \(\left[K_{w}: \mathbf{Q}_{v}\right]\), show that for \(\alpha \in K, \alpha \neq 0\), we have $$ \prod_{w \in M_{K}}|\alpha|_{w}^{N_{w}}=1 $$

Show that the \(p\) -adic numbers \(Q_{p}\) have no automorphisms other than the identity. [Hint: Show that such automorphisms are continuous for the \(p\) -adic topology. Use Corollary \(7.5\) as an algebraic characterization of elements close to \(1 .]\)

Let \(A\) be a principal entire ring, and let \(K\) be its quotient field. Let o be a valuation ring of \(K\) containing \(A\), and assume \(0 \neq K\). Show that o is the local ring \(A_{(D)}\) for some prime element \(p\). [This applies both to the ring \(\mathbf{Z}\) and to a polynomial ring \(k[X]\) over a field \(k\).]

Let \(Q_{p}\), be a \(p\) -adic field. Show that \(Q\), contains infinitely many quadratic fields of type \(\mathbf{Q}(\sqrt{-m})\), where \(m\) is a positive integer.

If \(K\) is a field complete with respect to a discrete valuation, with finite residue class field, and if 0 is the ring of elements of \(K\) whose orders are \(\geqq 0\), show that o is compact. Show that the group of units of o is closed in \(o\) and is compact.

Let \(K\) be a field complete with respect to a discrete valuation, let o be the ring of integers of \(K\), and assume that o is compact. Let \(f_{1}, f_{2}, \ldots\) be a sequence of polynomials in \(n\) variables, with coefficients in o. Assume that all these polynomials have degree \(\leqq d\), and that they converge to a polynomial \(f\) (i.e. that \(\left|f-f_{i}\right| \rightarrow 0\) as \(i \rightarrow \infty\) ). If each \(f_{i}\) has a zero in 0 , show that \(f\) has a zero in o. If the polynomials \(f_{i}\) are homogeneous of degree \(d\), and if each \(f_{1}\) has a non-trivial zero in o, show that \(f\) has a non-trivial zero in o. [Hint: Use the compactness of o and of the units of o for the homogeneous case.]

Show that if \(p, p^{\prime}\) are two distinet prime numbers, then the fields \(\mathbf{Q}_{p}\) and \(\mathbf{Q}_{\boldsymbol{r}}\) are not isomorphic.

Prove that the field \(\mathbf{Q}_{p}\) contains all \((p-1)\) -th roots of unity. [Hint : Use Proposition 7.6, applied to the polynomial \(X^{p-1}-1\) which splits into factors of degree 1 in the residue class field. ] Show that two distinct \((p-1)\) -th roots of unity cannot be congruent mod \(p\).

(a) Let \(f(X)\) be a polynomial of degree 1 in \(\mathbf{Z}[X]\). Show that the values \(f(a)\) for \(a \in \mathbf{Z}\) are divisible by infinitely many primes. (b) Let \(F\) be a finite extension of \(\mathbf{Q}\). Show that there are infinitely many primes \(p\) such that all conjugates of \(F\) (in an algebraic closure of \(\mathbf{Q}_{p}\) ) actually are contained in \(\mathbf{Q}_{p}\). [Hint: Use the irreducible polynomial of a generator for a Galois extension of \(\mathbf{Q}\) containing \(F .]\)

Let \(K\) be a field of characteristic 0 , complete with respect to a non- archimedean absolute value. Show that the series $$ \begin{aligned} \exp (x) &=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots \\ \log (1+x) &=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots \end{aligned} $$ converge in some neighborhood of \(0 .\) (The main problem arises when the characteristic of the residue class field is \(p>0\), so that \(p\) divides the denominators \(n !\) and \(n\). Get an expression which determines the power of \(p\) occurring in \(n !\) !) Prove that the exp and log give mappings inverse to each other, from a neighborhood of 0 to a neighborhood of 1 .

Let \(K\) be as in the preceding exercise, of characteristic 0, complete with respect to a nonarchimedean absolute value. For every integer \(n>0\), show that the usual binomial expansion for \((1+x)^{1 / n}\) converges in some neighborhood of \(0 .\) Do this first assuming that the characteristic of the residue class field does not divide \(n\), in which case the assertion is much simpler to prove.

Let \(F\) be a complete field with respect to a discrete valuation, let o be the valuation ring. \(\pi\) a prime element, and assume that \(0 /(\pi)=k\). Prove that if \(a, b \in 0\) and \(a \equiv b\left(\bmod \pi^{\prime}\right)\) with \(r>0\) then \(a^{r^{n}} \equiv b^{p^{n}}\left(\bmod \pi^{r+\pi}\right)\) for all integers \(n \geqq 0\).

(Local uniformization). Let \(k\) be a field, \(K\) a finitely generated extension of transcendence degree 1 , and o a discrete valuation ring of \(K\) over \(k\), with maximal ideal \(\mathrm{m}\). Assume that \(0 / \mathrm{m}=k\). Let \(x\) be a generator of \(\mathrm{m}\), and assume that \(K\) is separable over \(k(x)\). Show that there exists an element \(y \in 0\) such that \(K=k(x, y)\), and also having the following property. Let \(\varphi\) be the place on \(K\) determined by 0. Let \(a=\varphi(x), b=\varphi(y)\) (of course \(a=0\) ). Let \(f(X, Y)\) be the irreducible polynomual in \(k[X, Y]\) such that \(f(x, y)=0 .\) Then \(D_{2} f(a, b) \neq 0 .\) [Hint: Write first \(K=k(x, z)\) where \(z\) is integral over \(k[x]\). Let \(z=z_{1}, \ldots, z_{n}(n \geqq 2)\) be the conjugates of \(z\) over \(k(x)\), and extend \(o\) to a valuation ring \(\mathrm{O}\) of \(k\left(x, z_{1}, \ldots, z_{n}\right) .\) Let $$ z=a_{0}+a_{1} x+\cdots+a_{r} x^{r}+\cdots $$ be the power series expansion of \(z\) with \(a_{i} \in k\), and let \(P_{X}(x)=a_{0}+\cdots+a, x^{\prime} .\) For \(i=1, \ldots, n\) let $$ y_{i}=\frac{z_{i}-P_{r}(x)}{x^{\prime}} $$ Taking \(r\) large enough, show that \(y_{1}\) has no pole at \(D\) but \(y_{2}, \ldots, y_{n}\) have poles at \(D\). The elements \(y_{1}, \ldots, y_{n}\) are conjugate over \(k(x) .\) Let \(f(X, Y)\) be the irreducible polynomial of \((x, y)\) over \(k\). Then \(f(x, Y)=\psi_{n}(x) Y^{n}+\cdots+\psi_{0}(x)\) with \(\psi_{i}(x) k[x]\). We may also assume \(\psi_{1}(0) \neq 0\) (since \(f\) is irreducible). Write \(f(x, Y)\) in the form $$ f(x, Y)=\psi_{n}(x) y_{2} \cdots y_{n}\left(Y-y_{1}\right)\left(y_{2}^{-1} Y-1\right) \cdots\left(y_{n}^{-1} Y-1\right) $$ Show that \(\psi_{n}(x) y_{2} \cdots y_{n}=u\) does not have a pole at \(D .\) If \(w \in D\), let \(w\) denote its residue class modulo the maximal ideal of \(\mathrm{O}\). Then $$ 0 \neq f(\bar{x}, Y)=(-1)^{n-1} \bar{u}\left(Y-\bar{y}_{1}\right) $$ Let \(y=y_{1}, \bar{y}=b\). We find that \(\left.D_{2} f(a, b)=(-1)^{n-1} \bar{u} \neq 0 .\right]\)

(Iss'sa-Hironaka Ann. of Math 83 (1966), pp. 34-46). This exercise requires a good working knowledge of complex variables. Let \(K\) be the field of meromorphic functions on the complex plane C. Let \(D\) be a discrete valuation ring of \(K\) (containing the constants C). Show that the function \(z\) is in \(\emptyset\). [Hint: Let \(a_{1}, a_{2}, \ldots\) be a discrete sequence of complex numbers tending to infinity, for instance the positive integers. Let \(v_{1}, v_{2}, \ldots\), be a sequence of integers, \(0 \leqq v_{i} \leqq p-1\), for some prime number \(p\), such that \(\sum v_{i} p^{i}\) is not the \(p\) -adic expansion of a rational number. Let \(f\) be an entire function having a zero of order \(v_{i} p^{i}\) at \(a_{i}\) for each \(i\) and no other zero. If \(z\) is not in o, consider the quotient $$ g(z)=\frac{f(z)}{\prod_{i=1}^{n}\left(z-a_{i}\right)^{v_{i} p^{\prime}}} $$ From the Weierstrass factorization of an entire function, show that \(g(z)=h(z)^{p^{n+1}}\) for some entire function \(h(z)\). Now analyze the zero of \(g\) at the discrete valuation of o in terms of that of \(f\) and \(\prod\left(z-a_{i}\right)^{v_{i} p^{t}}\) to get a contradiction. \(]\) If \(U\) is a non-compact Riemann surface, and \(L\) is the field of meromorphic functions on \(U\), and if \(o\) is a discrete valuation ring of \(L\) containing the constants, show that every holomorphic function \(\varphi\) on \(U\) lies in o. [Hint: Map \(\varphi: U \rightarrow \mathbf{C}\), and get a discrete valuation of \(K\) by composing \(\varphi\) with meromorphic functions on C. Apply the first part of the exercise.] Show that the valuation ring is the one associated with a complex number. [Further hint: If you don't know about Riemann surfaces, do it for the complex plane. For each \(z \in U\), let \(f_{z}\) be a function holomorphic on \(U\) and having only a zero of order 1 at \(z\). If for some \(z_{0}\) the function \(f_{x 0}\) has order \(\geqq 1\) at 0, then show that 0 is the valuation ring associated with \(z_{0}\). Otherwise, every function \(f_{z}\), has order 0 at o. Conclude that the valuation of o is trivial on any holomorphic function by a limit trick analogous to that of the first part of the exercise.]