By Dummit D. S

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21. (a) Under the hypothesis of the theorem X n − 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 ∈ / −4k 4 . See Lang, Algebra, VIII, §9, Theorem 16. (b) If F has characteristic p (hence has no pth roots of 1 other than 1), then X p − 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.

FIELDS AND GALOIS THEORY 47 Proof. We deﬁne an ordering on the monomials in the Xi by requiring that X1i1 X2i2 · · · Xnin > X1j1 X2j2 · · · Xnjn if either i1 + i2 + · · · + in > j1 + j2 + · · · + jn or equality holds and, for some s, i1 = j1 , . . , is = js , but is+1 > js+1 . For example, X1 X23 X3 > X1 X22 X3 > X1 X2 X32 . Let X1k1 · · · Xnkn be the highest monomial occurring in P with a coeﬃcient c = 0. Because P is symmetric, it contains all monomials obtained from X1k1 · · · Xnkn by permuting the X’s.

Then F contains one ﬁeld Fpn for each integer n ≥ 1— n it consists of all roots of X p − X—and Fpm ⊂ Fpn ⇐⇒ m|n. The partially ordered set of ﬁnite subﬁelds of F is isomorphic to the set of integers n ≥ 1 partially ordered by divisibility. Finite ﬁelds were sometimes called Galois ﬁelds, and Fq used to be denoted GF (q) (it still is in Maple). Maple contains a “Galois ﬁeld package” to do computations in ﬁnite ﬁelds. , a generator for F× q ). To start it, type: readlib(GF);. 7. Computing Galois groups over Q.