Exercise 10, § 8

Commutative Algebra, VI, 464

Let w be the 2-adic valuation on the 2-adic field $\mathbf{Q}_2$ ; on the field of rational functions $\mathbf{Q}_2(X)$ consider the discrete valuation $v$ extending $w$ and such that

v(a_0 + a_1 X + \cdots + a_n X^n) = \inf_i (w(a_i))

(§ 10, no. 1, Lemma 1); let K be the completion of

\mathbf{Q}_2(X)

with respect to this valuation and let its valuation also be denoted by

v

. Let L be the splitting field of the polynomial

(Y^2 - X)^2 - 2

K[Y]

and let

v'

be the unique valuation on L extending

v

. Show that

[L : K] = 8, e(v'/v) = 4, f(v'/v) = 2

; if k (resp.

k'

) is the residue field of

v

(resp.

v'

k'

is a radicial extension of k of degree 2; show that there exists no sub-extension E of L such that

[E : K] = 2

for which the residue field of the restriction of

v'

to E is equal to

k'

. (Of necessity

E = K(\sqrt{\alpha})

, where

\alpha \in K

would not be a square in K but such that

\alpha \equiv X \pmod{2}

; express &with the aid of a suitable basis of L over

K(\sqrt{2})

and observe that there exists in

K(\sqrt{2})

no element whose square is congruent to X mod. 2.)

Solution

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