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**spudwish** Yes. Given any polynomial, say, over Q, we know it splits into linear factors "somewhere". So, we can enumerate the roots, in whatever fashion we like, so that embeddings correspond to permutations of {1,...,n}. The problem with this using this perspective in computing the Galois group G is that we might only find G up to conjugacy.

From what books did you learn Galois theory? I've read a few books on the subject and all but one (Lang's Algebra) take the permutation approach.