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Unformatted text preview: Math 5126 Monday, March 10 Sixth Homework Solutions 1. Section 13.4, Exercise 3 on page 545. Determine the splitting field and its degree over Q for x 4 + x 2 + 1. x 4 + x 2 + 1 = ( x 2 + x + 1 )( x 2- x + 1 ) , so the roots are 1 - 3 2 . Therefore the splitting field is contained in Q ( - 3 ) . This has degree 2 over Q and since the splitting field is clearly not Q , we conclude that the splitting field is Q ( - 3 ) and that it has degree 2 over Q . 2. Let f denote the minimal polynomial of a over K and let n denote the largest nonneg- ative integer such that f is a polynomial in x p n . Then we may write f = g ( x p n ) for some polynomial g whose derivative is nonzero. Note that g is irreducible because f is, and g is separable because also its derivative is nonzero. Since a p n satisfies g , this proves that a is separable. 3. Since D ( x 2 n + 1 ) = x 2 n and D ( x 2 n ) = 0, we see that Df is a sum of elements of the form x 2 n . Since x 2 n = ( x n ) 2 , we see that...
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