MAT444Dec08_MAT444Final

MAT444Dec08_MAT444Final - Math 442 Final Examination and...

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: Math 442 Final Examination and Qualifying Examination in Linear Algebra and Field Theory December, 2008 Write your answer to each problem on a separate sheet, placing the problem number in the upper right corner. Be sure to justify all your answers. Do not cite any result that reduces a proof to a triviality. Field Theory 1. [5,5,5,8] Let K be a field, let f ( X ) be a polynomial in K [ X ] and suppose deg( f ) = n . a. Prove that every element of K [ X ] / h f ( X ) i has a unique representation h ( X )+ h f ( X ) i , where deg( h ) < n . b. Explain in detail why f ( X ) must be irreducible in K [ X ] in order for K [ X ] / h f ( X ) i to be a field. c. Suppose that f ( X ) is irreducible in K [ X ]. Prove that K [ X ] / h f ( X ) i = K ( ), where satisfies f ( ) = 0. d. Suppose that f ( X ) and g ( X ) are irreducible polynomials over the field K and that the greatest common divisor gcd(deg( f ) , deg( g )) = 1. Let be a zero of f in some algebraic closure of K . Show that....
View Full Document

Page1 / 2

MAT444Dec08_MAT444Final - Math 442 Final Examination and...

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