Galois theory and algorithms Steven Weintraub's book {\em A Guide to Advanced Linear Algebra} includes the following remark:
"Of course, there is no algorithm for factoring polynomials, as we know from Galois theory."
I can't make sense of this.  I feel confident that Galois theory doesn't speak to the question of algorithms, and confident that there do exist algorithms for factoring integer polynomials over the integers (after Kronecker), and strategies for computing in the field of algebraic numbers that make tautological the question of factoring polynomials irreducible over the rationals.
Have I missed some way to salvage this remark?
 A: When I was a graduate student, I fell into conversation with a professor from another university who was chortling over a paper that presented an algorithm for solving quintic polynomials.  (The algorithm did not purport to find a solution in radicals.)  He was sure the author must be a fool for not realizing that in light of Galois, there can be no such algorithm.  When I expressed something like surprise/doubt, it quickly became apparent that this full professor of mathematics didn't have a clue what Galois had or had not proved --- and had no interest in finding out.  But he persisted in his belief that "by Galois" there can be no such algorithm.
My guess is that Weintraub (like all of us from time to time) made a mental slip from which he (unlike my professor friend) would immediately retreat if it were pointed out.
This probably should have been a comment but it was too long to fit.
A: The roots of any polynomial with complex coefficients can in fact be given in terms of generalized hypergeometric functions or theta constants. There is an appendix of Mumford's "Lectures on Theta" by Umemura, where the later is explained in some detail. On the other hand the Jordan canonical form is numerically unstable, so that description is not really useful. 
A: Hermite ca 1858 in Comptes Rendus, solved the general 5-ic using the j-function
and the modular polynomial Phi_5(j(z), j(5z)). 
A: As Noam Elkies wrote, this statement was made in the context of finding eigenvalues. The full paragraph reads "We not only show the existence of the Jordan canonical form, but also derive an algorithm for finding the Jordan canonical form of T as well as finding a Jordan basis of V, assuming we can factor the characteristic polynomial c_T(x). (Of course, there is no algorithm for factoring polynomials, as we know from Galois theory.)" Thus, from the context it is clear that what was meant by factoring was factoring into a product of linear factors over an arbitrary field F.
A: You are absolutely correct, this statement as stated does not make much sense.  Over the integers (or any algebraic extension thereof), there are known algorithms for factoring multivariate polynomials.  Any textbook on Computer Algebra will list some of them.
This has been an area of research with ups and downs, with a recent resurgence.  The work of Mark van Hoeij (scroll down to the section on polynomials) is especially impressive.  He has the fastest currently known algorithms, in theory and in practice, for the problem.
There are even algorithms for absolute factorization, i.e. finding the exact algebraic extension needed of the base field for a (univariate) polynomial to split into linear factors.  Most CASes have implementations (see ?evala,AFactor in Maple, for example).
