# ahw6 - Math 5126 Monday March 10 Sixth Homework Solutions 1...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

ahw6 - Math 5126 Monday March 10 Sixth Homework Solutions 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online