MATH-GA 2210.001: Homework Local Fields 1

1. Determine all the absolute values for the following fields:

(a) k =C, (b) k =R,

(c) k =Fq a finite field with q=p^{r} elements.

2. Let | |_{p}_{1},| |_{p}_{2}, . . .| |_{p}_{k} be nontrivial inequivalent absolute values on Q corre-
sponding to distinct primesp_{i},i= 1, . . . k, and let a_{1}, . . . a_{k} be elements ofQ.
Let d be the common denominator of a_{i}. Show that for every >0 there is
an elementa ∈K such that |a−a_{i}|_{p}_{i} < for i= 1, . . . n and |a|_{p} <1/|d| for
all absolute values corresponding to primes pdistinct form p_{i}, i= 1, . . . k.

3. Let k be a field and let a_{1}, . . . a_{n} (resp. b_{1}, . . . b_{n}) be distinct elements of k.

LetK =k(t) a purely transcendental extension of k. Show that there exists
x∈Ksuch that the functionsx−b_{i}have a simple zero att=a_{i}fori= 1, . . . n.

4. LetL=C(x)[p

x(x−1)(x+ 1)]be a degree2extension of a purely transcen-
dental extensionC(x)ofC, generated byywithy^{2} =x(x−1)(x+1). The goal
of this exercise is to show thatLis not isomorphic to a purely transcendental
extensionC(t) of C.

(a) Let v :L^{∗} →Z be a valuation onL. Show that v(x) is even.

(b) Show that x is not a square in L.

(c) Let z ∈C(t)be an element, such that for any valuation
v :C(t)^{∗} →Z,

one has that v(z) is even. Show that z is a square in C(t) (Hint: if z is divisible by t−α, forα ∈C, consider the valuationv given by the order of vanishing at α.)

(d) Conclude that L is not isomorphic to a purely transcendental extension C(t) of C.

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