Does anyone know what are the most efficient algorithms for factoring polynomials in a field of characteristic zero, i.e, a field that may contain infinitely many elements. I'm mainly concerned within the context of the field of integers but I wouldn't mind the rationals as well.
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$\begingroup$ What research have you done? Wikipedia has an article on this; have you read it? Have you followed up on its references? In the future I'd expect you to do more research, and to show us what research you've done in the question: if your question is partly or wholly answered by Wikipedia and you were unaware, you haven't done enough research before asking. $\endgroup$– D.W. ♦Jul 18 '14 at 6:04
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$\begingroup$ Also, please edit your question to clarify what you are looking for (since the integers are not a field; they are a ring). Also, are you looking at univariate polynomials or multivariate polynomials? $\endgroup$– D.W. ♦Jul 18 '14 at 6:05
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Do you know of Berlekamp's algorithm and the Cantor–Zassenhaus algorithm?
http://en.wikipedia.org/wiki/Berlekamp%27s_algorithm
http://en.wikipedia.org/wiki/Cantor%E2%80%93Zassenhaus_algorithm
Unless I'm misunderstanding your question, there is no field of integers- they form a ring but don't have multiplicative inverses
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1$\begingroup$ Answers that only contain links are not very useful (permanently). Please add some value! $\endgroup$– Raphael ♦Jul 17 '14 at 20:49