f09_mthsc852_hw15

# f09_mthsc852_hw15 - L is normal over F and Gal L/F is...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 15 Due Monday, October 19, 2009 (1) (a) Find a primitive element over Q for K = Q ( 3 , 3 2) C . (b) Find a primitive element over Q for a splitting ﬁeld K C for the polynomial f ( x ) = x 4 - 5 x 2 + 6. (2) Let K/F be a normal ﬁeld extension and f ( x ) F [ x ] an irreducible polynomial over F . (a) Prove that if f ( x ) splits in K , all zeros of f ( x ) in K have the same multiplicity. (a) Now suppose that f ( x ) does not split in K . Prove that all irreducible factors of f ( x ) in K [ x ] have the same degree. (3) (a) Let F be a ﬁeld of characteristic zero, and let p be a prime such that p | [ K : F ] for every ﬁeld extension K/F of ﬁnite degree. Prove that [ K : F ] is a power of p whenever K/F is an extension of ﬁnite degree. (b) Let F be a ﬁeld, Q F A , maximal with respect to 2 6∈ F (Why does F exist?). (i) If F K A , with K normal and ﬁnite over F , and K 6 = F , show that G = Gal( K/F ) is a 2-group having a unique subgroup of index 2. Conclude that G is cyclic. (ii) If F L A and [ L : F ] is ﬁnite show that
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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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