Adv Alegbra HW Solutions 123

Adv Alegbra HW Solutions 123 - 123 3.101 Generalize...

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Unformatted text preview: 123 3.101 Generalize Proposition 3.116(iii) as follows. Let : k k be an iso- morphism of fi elds, let E / k and E / k be extensions, let p ( x ) k [ x ] and p ( x ) k [ x ] be irreducible polynomials (as in Theorem 3.33, if p ( x ) = a i x i , then p ( x ) = ( a i ) x i ), and let z E and z E be roots of p ( x ), p ( x ) , respectively. Then there exists an isomorphism e : k ( z ) k ( z ) with e ( z ) = z and with e extending . k ( z ) e / k ( z ) k / k Solution. Absent. 3.102 Let f ( x ) = a + a 1 x + + a n 1 x n 1 + x n k [ x ] , where k is a fi eld, and suppose that f ( x ) = ( x r 1 )( x r 2 ) . . . ( x r n ) E [ x ] , where E is some fi eld containing k . Prove that a n 1 = ( r 1 + r 2 + + r n ) and a = ( 1 ) n r 1 r 2 r n . Conclude that the sum and the product of all the roots of f ( x ) lie in k ....
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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