This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
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.
 Fall '07
 PALinnell
 Math, Algebra

Click to edit the document details