having a hard time motivating or picturing myself using it on a daily basis.

>>16711975 (OP)
>Galois/ring theory
Start with just ring theory or group theory. Galois theory comes much later. There is an active thread >>16706377 with textbook recommendations. Ring theory is the most intuitive to start with imo because
>[math]\mathbb Z[/math] is a ring
>[math]\mathbb Q[/math] is a ring
>[math]\mathbb R[/math] is a ring
>[math]\mathbb C[/math] is a ring
>[math]\mathbb Z[x][/math] i.e. polynomials with coefficients in [math]\mathbb Z[/math], is a ring
>…and similarly for Q,R,C
>[math]\mathbb Z / \langle n \rangle[/math] i.e. the integers mod n, is a ring
So rings are just generally intuitive since you have seen so many of them before
>picture myself using Galois theory on a daily basis
Lol..?

ive taken group theory as a req course, but i had a hard time moving to group theory. thanks.

moving to galois* theory

>>16711975 (OP)
>Ayyyy lmao

>>16711996
Galois theory is motivated by questions about elements of [math]\mathbb C[/math], namely: if I have [math]z_1[/math] then can I get [math]z_2[/math] by some kind of simple operations. E.g. I can get [math]2z_1^2+1[/math] pretty easily with just addition and multiplication, and we could get [math]\sqrt {z_1} [/math] but this requires a somewhat weirder operation that maybe we don’t want to allow.

In practice the operations we are looking at might be compass-and-straightedge constructions, in which case square roots should be ok (construct a right triangle and look at the length of its hypotenuse) but then maybe cube roots could be a problem. Rule systems like this get formalized into something about polynomials f in [math]\mathbb C[z_1][/math] i.e. z_1 can be used in the coefficients, and then we’re trying to make z_2 be one of the roots.

So some of these questions can be answered by looking more abstractly at the relationship between z_1, and the possible roots z_2 of its polynomials [math]\mathbb C[z_1][/math]. What comes out of it is Galois’ theorem, which is one of the really beautiful theorems out there, and then some straightforward answers to old questions like, can you trisect an angle using compass and straightedge, or can you solve a quintic equation by radicals. There isn’t much you can do past that point I don’t think; it’s sort of a happy ending to the “main quest” in the abstract-algebra subplot. Worth your time imo,

>>16712010
>[math]\mathbb C[z_1][/math]
I guess I mean [math]\mathbb C(z_1)[x][/math], i.e. the coefficients can use z_1 also in the denominator and the polynomials have a variable c. It’s useful to look at ring theory and field theory a bit first just to get this notation down pat.

>>16712011
>variable c
I mean x…

>>16712011
I mean [math]\mathbb Q(z_1)[x][/math]…

>>16711975 (OP)
>having a hard time motivating or picturing myself using it on a daily basis
If you're asking this question at all there is no reason to do any math past third grade.