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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.


Solution Summary

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