ahw8 - Math 5126 Monday March 24 Eighth Homework Solutions...

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Unformatted text preview: Math 5126 Monday, March 24 Eighth Homework Solutions 1. Write F- = { a 1 ,..., a q } , where q + 1 = | F | . Then x q- 1 = ( x- a 1 )( x- a 2 ) ... ( x- a q ) . The coefficient of x q- 1 on the right hand side is- ( a 1 + ··· + a q ) , whereas the coefficient of x q- 1 on the left hand side is 0 (because q ≥ 3). This shows that the sum of all elements of F is zero. Next consider the constant coefficient. We obtain- 1 = (- 1 ) q- 1 a 1 a 2 ... a q . Note that (- 1 ) q- 1 = 1, because if not, then q is even which means that F has characteristic 2 and then- 1 = 1. Thus in all cases- 1 = a 1 a 2 ... a q . Finally the sum of all products of pairs of elements of F is the square of the sum of the elements of F and hence by the first part is zero. 2. Suppose there are a finite number of subfields between F and K . Then we may write K = F ( α ) for some α ∈ K . Let f denote the minimal polynomial of α over K , a polynomial of degree 8, and let E be a splitting field for f over F containing...
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This note was uploaded on 01/29/2009 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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ahw8 - Math 5126 Monday March 24 Eighth Homework Solutions...

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