I've been reading some history and some background to Galois theory and these usually talk about the "group of an equation". I'm assuming this is the same as the "Galois group" which I've been taught. However, the galois group, as i know it, is a group of embeddings (injective ring homomorphism) under composition, whereas the "group of an equation" is always talked about as being a group of permutations of roots. I'm suspect these two things to be the same and I'm trying to see the connection. Could anyone say if the follow is correct or if I'm on the right lines?
Let be a field. Let be an irreducible polynomial in . Let be a root of in some field extension .
So the definition of the Galois group is .
But we know each K embedding is uniquely defined by its behaviour on the root . Futhermore, Artin's Extension theorem, implies that any K embedding will send to another root in the splitting field of over . (note that beta may equal alpha). If, infact it turns out that is actually in . Then this particular K-embedding (sending alpha to beta) is in the Galois group.
So in this way, each K embedding corresponds to a permutation of a root of . i.e. alpha being sent to beta.
So the Galois group of K embeddings from L to L is equivalent to a, particular, subgroup of the group of permutation of f's roots.
Is this above correct?
Thanks for any clarification!