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Irreducuble Polynomials : Find the Minimum Polynomial and Factorization

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).

(See attached file for full problem description with proper symbols)

The field in Q, the rational numbers.

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Solution Summary

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

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