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

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Math 5126 Monday, March 24 Eighth Homework Solutions 1. Write F - 0 = { 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 K . Let a 1 = α , a 2 ,..., a 8 be the roots of f in E (there maybe repetitions). Let L be an intermediate subfield, that is F L
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