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Irreducuble Polynomials : Find the Minimum Polynomial and Factorization

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Find (with proof) an element a Є Q such that Q(√3, √10)= Q(a) . Find the minimum polynomial of √3 + √10 . Hence, or otherwise, show that x^4 -26x^2 + 49 is irreducible over Q.
Find the minimum polynomial of √3 + √10 over Q(√3) . Hence or otherwise factorize x^4 -26x^2 + 49 as a product of quadratic factors over Q(√3).

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The field in Q, the rational numbers.

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Minimum polynomials and factorization are investigated. The solution is detailed and well presented.

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1. Find , such that
Let . Since , then we have . Since , then we have . ...

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