5.6. Cyclic extensions.

Back

We are now able to classify the cyclic extensions of degree n of a field F in the case that F

contains n nth roots of 1.

Theorem 5.20. Let F be a field containing a primitive nth root of 1.

(a) The Galois group of Xn − a is cyclic of order dividing n.

(b) Conversely, if E is cyclic of degree n over F, then there is an element β ∈ E such that

E = F[β] and b =df βn ∈ F; hence E is the splitting field of Xn − b.

Proof. (a) If α is one root of Xn − a, then the other roots are the elements of the form

ζα with ζ an n th root of 1. Hence the splitting field of Xn − a is F[α]. The map σ → σα

α is

an injective homomorphism of Gal(F[α]/F ) into the cyclic group <ζ> .

(b) Let ζ be a primitive nth root of 1 in F, and let σ generate Gal(E/F). Then Nmζ =

ζn = 1, and so, according to Hilbert’s Theorem 90, there is an element β ∈ E such that

σβ = ζβ. Then σiβ = ζiβ, and so only the identity element of Gal(E/F[β]) fixes β—we

conclude by the Fundamental Theorem of Galois Theory that E = F[β]. On the other hand

σβn = ζnβn = βn, and so βn ∈ F.

Remark 5.21. (a) Under the hypothesis of the theorem Xn − a is irreducible, and its

Galois group is of order n, if

(i) a is not a pth power for any p dividing n;

(ii) if 4|n then a /∈ −4k4.

See Lang, Algebra, VIII, §9, Theorem 16.

(b) If F has characteristic p (hence has no pth roots of 1 other than 1), then Xp − X − a

is irreducible in F[X] unless a = bp −b for some b ∈ F, and when it is irreducible, its Galois

group is cyclic of order p (generated by α → α + 1 where α is a root). Moreover, every

extension of F which is cyclic of degree p is the splitting field of such a polynomial.

Remark 5.22 (Kummer theory). Above we gave a description of all Galois extensions of

F with Galois group cyclic of order n in the case that F contains a primitive nth root of

1. Under the same assumption on F, it is possible to give a description of all the Galois

extensions of F with abelian Galois group of exponent n, i.e., a quotient of (Z/nZ)r for some

r.

Let E be such an extension of F, and let

S(E) = {a ∈ F× | a becomes an nth power in E};

Then S(E) is a subgroup of F× containing F×n, and the map E → S(E) defines a oneto-

one correspondence between abelian extensions of E of exponent n and groups S(E),

F× ⊃ S(E) ⊃ F×n, such that (S(E) : F×n) < ∞. The field E is recovered from S(E) as the

splitting field of

_

(Xn−a) (product over a set of representatives for S(E)/F ×n). Moreover,

there is a perfect pairing

(a, σ) → σa

a

:

S(E)

F×n

× Gal(E/F) → μn (group of nth roots of 1).

In particular, [E : F] = (S(E) : F×n). (Cf. Exercise 5 for the case n = 2.)

FIELDS AND GALOIS THEORY 45

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