Unformatted text preview: L is normal over F and Gal( L/F ) is cyclic. Conclude that the set of ﬁnite extensions of F (in A ) is an ascending chain. (4) Suppose K/F has ﬁnite degree and char F-[ K : F ]. Show that K/F is separable. (5) Let F be a ﬁeld of characteristic p and f ( x ) = x p-a ∈ F [ x ]. Prove that f ( x ) is either irreducible in F [ x ] or splits in F [ x ]. (6) Suppose char F = p 6 = 0 and K is an extension of F . An element a ∈ K is called purely inseparable over F if it is a root of a polynomial of the form x p k-b ∈ F [ x ], 0 ≤ k ∈ Z . (a) Show that if a ∈ K is both separable and purely inseparable over F , then a ∈ F . (b) Show that the set of all elements of K that are purely inseparable over F constitute a ﬁeld. Conclude that there is a unique largest “purely inseparable” extension of F within K ....
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- Fall '08
- Algebra, Complex number, finite field, 0 K, finite degree, Dr. Matthew Macauley