automorphisms

The easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.

So, what physics means is that translations are essentially automorphisms.

Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms.

A representation of a group, formally, is a homomorphism of that group to the group of automorphisms of some vector space.

Most of the time my work involves reasoning about group actions, automorphisms, arc-stabilisers, etc.

The first example that comes to mind is the question of whether all automorphisms of the Calkin algebra are inner (Farah, 2011).

In particular, the core disagreement seems to be about whether the automorphisms of C should keep R (as a subset) fixed, or not.

I'm much more used to seeing it used for a dual space than a group of automorphisms.

PP meant automorphisms, which is what the OP article is about.

They aren't really special except that adjoining the solutions to a radical to a field make the associated group of automorphisms simplify in a nice way.

Math is full of the novel (at the time of invention) concepts: complex numbers, groups, automorphisms, etc., etc.

Algebraic number theory, in particular Galois theory, studies field extensions by looking at the group of symmetries: the field automorphisms of the larger field that fix the smaller field.

It's interesting that algebraic closure is enough to get you this far, but with the arbitrary choice of cardinality and all these "wild automorphisms", doesn't this construction just seem...

For example, the problem with understanding the notation "Aut(L|k)" is not the notation, because if the author wrote "group of field automorphisms of L fixing k elementwise" I'd be equally lost if I don't understand those words.

Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.

To me a forum like this is like when I'd be sitting around in uni wondering aloud about some problem and someone will say "Well, maybe if you use the fact that the functions which invert the sign of the √2 and √3 parts in Q(√2, √3) are automorphisms, then you can see that the group containing them and their composition is just Z2×Z2".