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Unformatted text preview: Abstract Algebra II Spring 2010 Final Exam - Solutions Solve any 6 of these 8 problems. If you solve more than 6 problems I will grade all of them and will take into account 6 best scores. Each problem is worth 16 points. Everyone gets 4 points for signing their paper. Problem 1. Find the number of irreducible polynomials of the form x 2 + ax + b over the field F p for a prime p . Lets find the number of reducible polynomials of this form. Such polynomials would factor as ( x- )( x- ). There are ( p 2 ) ways to pick such distinct and and p ways to pick and if they coincide. Hence there are p ( p- 1) / 2 + p reducible polynomials of the form x 2 + ax + b and therefore the number of irreducible once is p 2- p ( p- 1) / 2- p = p ( p- 1) / 2. Alternative solution: We want to compute the number of all monic irreducible polynomi- als of degree 2. We had proved that x p 2- 1 is the product of all the irreducible polynomials of degree 1 and 2. The number of irreducible polynomials of degree 1 is p . Hence the product of all irreducible polynomials of degree 2 has degree p 2- p and hence the number of such polynomials is ( p 2- p ) / 2....
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