solutions04

solutions04 - Problem Set #4 Solutions Chapter III. 5.1.i....

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem Set #4 Solutions Chapter III. 5.1.i. Write out the elementary symmetric polynomials for n = 3 , 4 , 5 . Using the form given on p. 103, k = X 1 i 1 <i 2 < <i k n x i 1 x i 2 x i k , we do the case n = 5: 1 = x 1 + x 2 + x 3 + x 4 + x 5 2 = x 1 x 2 + x 1 x 3 + x 1 x 4 + x 1 x 5 + x 2 x 3 + x 2 x 4 + x 2 x 5 + x 3 x 4 + x 3 x 5 + x 4 x 5 3 = x 1 x 2 x 3 + x 1 x 2 x 4 + x 1 x 2 x 5 + x 1 x 3 x 4 + x 1 x 3 x 5 + x 1 x 4 x 5 + x 2 x 3 x 4 + x 2 x 3 x 5 + x 2 x 4 x 5 + x 3 x 4 x 5 4 = x 1 x 2 x 3 x 4 + x 1 x 2 x 3 x 5 + x 1 x 2 x 4 x 5 + x 1 x 3 x 4 x 5 + x 2 x 3 x 4 x 5 5 = x 1 x 2 x 3 x 4 x 5 . ii. How many monomial summands are there in k ? The number of monomial summands equals the number of ways of choosing k elements out of n elements, i.e., n k . 5.2. The general polynomial f ( x ) = Q n i =1 ( x- x i ) is irreducible in F [ x ] . Suppose f ( x ) = g ( x ) h ( x ) in F [ x ], with g ( x ) = Y i S ( x- x i ) , h ( x ) = Y i S c ( x- x i ) F [ x ] for some subset S { 1 , 2 ,...,n } . Supposing 1 S , 2 6 S , extend the F- automorphism = (12) S n of K to an F-automorphism of K [ x ] by setting ( x ) = x . By the preceeding paragraph in Grove, F = F S n , so fixes all elements of F [ x ]. But g ( x ) = Y i S ( x- x i ) = Y i S ( ( x )- x ( i ) ) = Y i S ( x- x ( i ) ) 6 = Y i S ( x- x i ) = g ( x ) , so g ( x ) 6 F [ x ], a contradiction....
View Full Document

This note was uploaded on 08/06/2008 for the course MATH 220 taught by Professor Morrison during the Spring '08 term at UCSB.

Page1 / 4

solutions04 - Problem Set #4 Solutions Chapter III. 5.1.i....

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online