Problem Set 6. Algebra - Problem Seminar Fall 2013 Problem...

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Problem Seminar. Fall 2013. Problem Set 6. Algebra. Classical results. 1. Vandermonde. Let V = 1 1 · · · 1 x 1 x 2 · · · x n x 2 1 x 2 2 · · · x 2 n . . . . . . . . . . . . x n - 1 1 x n - 1 2 · · · x n - 1 n . Then det( V ) = Q 1 i<j n ( x j - x i ) . 2. Cayley-Hamilton. Given an n × n matrix A the characteristic polynomial of A is defined as P A ( λ ) = det( λI n - A ) , where I n is the n × n identity matrix. Then P A ( A ) = 0 for every A. 3. Lagrange interpolation. For every positive integer n and every collection of real num- bers a 1 , a 2 , . . . , a n there exists a polynomial of degree at most n so that P (1) = a 1 , P (2) = a 2 , . . . , P ( n ) = a n . 4. Eisenstein criterion. Let P ( x ) = a n x n + a n - 1 x n - 1 + . . . + a 1 x + a 0 be a polynomial with integer coefficients and p be a prime so that (i) p divides a 0 , a 1 , . . . a n - 1 ; (ii) p does not divide a n ; (iii) p 2 does not divide a 0 . Then P ( x ) can not be expressed as a product of two non-constant polynomials with inte- ger coefficients.
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