Factoring Over Algebraic Extensions
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We can factor a polynomial, where is a finite field extension of . First, using square-free factorization, we may suppose that the polynomial is square-free. Next we write explicitly as an algebra over . We next pick a random element . By the primitive element theorem, generates over with high probability. If this is the case, we can compute the minimal polynomial, of over . Factoring
over, we determine that
(notice that is a reduced ring since is square-free), where corresponds to the element . Note that this is the unique decomposition of as a product fields. Hence this decomposition is the same as
where
is the factorization of over . By writing and generators of as a polynomials in, we can determine the embeddings of and into the components . By finding the minimal polynomial of in this ring, we have computed, and thus factored over
Read more about this topic: Factorization Of Polynomials
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