Question

does anyone know about Galois theory?

Answer 1

what do you want to know?

Answer 2

let's start with the definition and uses of a Galois group and the connection between polynomials and finite fields

Answer 3

You know they write books about this stuff?

Answer 4

Do you know what a normal group is?

Answer 5

yes I know what a normal group is. how does this relate to my question?

Answer 6

nothing, I was going to see if you could help me with something, but I figured it out.

Your question is basically, "explain all of Galios theory."

I would get a book and read it. You will not learn much about it here.

If you want to know the definition, google it. Else I am just going to write the same thing.

I could answer some questions, but I am not just going to explain Galios Theory when there are books for that.

Answer 7

I will tell you that Galios was the man. He died in a fight about a girl at age 21, and the night before he wrote up some of the most important mathematics in history

Answer 8

of course i know that there are books on this, and I don't want you to explain all of Galois theory. what I want to know is the relation between polynomials and finite fields.

Answer 9

that seems to be an important part of Galois theory. i.e. being able to find the zeros of a polynomial, or being able to determine whether a polynomial is irreducible

Answer 10

yes basically, if you have a polynomial over some field that has no roots, then you can throw in that root to the field and "close it up" and now it contains the root. But it might not contain all roots. So you can play the game again. Finally we look at all the fields inbetween.

Answer 11

Also, I am reading a book on graduate level abstract algebra.

Answer 12

Good deal. I TA for graduate level AA class.

Answer 13

well I did, it ended yesterday

Answer 14

I have done a lot of research on prime state algebraic logic, and there seems to by a direct connection. Also, i found a counter example of Fermat's last theorem in base 3:
\[x^3+y^3=x^3+3x^2y+3xy^2+y^3=(x+y)^3\]

Answer 15

ok so take some field extension E over some field F, The set of all automorphisms that fix elements in F is called the Galios group. Now if you look at all of the subgroups of that group, each one is inbetween field extensions. !!!

Answer 16

it's not FLT in mod 3....

Answer 17

So back to Galios (as you asked about) so if you take some field F and extend it to E, and make a latice of all the field extensions from F to E, it will look exactly like the upside down lattice for the subgroups of the Galios group. Super cool

Answer 18

I see. can you draw an example?

Answer 19

it will look exactly like any subgroup lattice.

Answer 20

ok, I am pretty familiar with that.

Answer 21

Gal\((\dfrac{\mathbb{C}}{\mathbb{R}})\) has two elements, the identity and the complex conjigate map. Note that each map fixes the reals. So the Galios group is \(\mathbb{Z}_2\) which has no proper non-trivial subgroups, so what we get out of this, is that there are no field extensions of \(\mathbb{R}\) that live properly between \(\mathbb{R}\) and \(\mathbb{C}\). Does this make sense?

Answer 22

yes. what about Gal\((\frac{R}{Q})\)?

Answer 23

that is harder

Answer 24

you need to convince yourself that the only automorphism that fixes Q is the identity

Answer 25

so the difference is that the complex conjugate of an element of \(C\) fixes the real part of the element?

Answer 26

what about Gal\((\frac{Q}{Z_5})\)?

Answer 27

lol how long we going to do this?

Answer 28

I want to understand galois groups, field extensions, and I was leading to the analysis of finite fields and algebraic logic

Answer 29

but last night you wanted to understand algebraic topology

Answer 30

put all of that together and you have a PHD

Answer 31

yes, and I have been working on my thesis, but I am still an undergraduate at De Anza in Cupertino. Furthermore, I am currently working on understanding and proving the Hodge conjecture

Answer 32

Yeah, I do
Depends on what question you want to apply it on

Answer 33

https://en.wikipedia.org/wiki/Galois_theory

Answer 34

https://en.wikipedia.org/wiki/Hodge_conjecture

Answer 35

lol

Answer 36

yeah, I guess wikipedia is a good place to start :P

Answer 37

did you bump this?

Answer 38

yes

Answer 39

i decided that I would bump it before closing it

Answer 40

if no one was interested, i would close it and ask something else

Answer 41

k it's just us... am closing