Message from @ThisIsChris

Discord ID: 503833841392156673


2018-10-22 07:28:20 UTC  

which means they can't be isomorphic

2018-10-22 07:28:42 UTC  

the Galois fields can't be isomorphic that is

2018-10-22 07:29:51 UTC  

Hm I think you can actually prove it if the degrees mismatch too. Since you assume the rep g has degree strictly less than f1, then g*f2/f1 should have degree strictly less than f2

2018-10-22 07:30:43 UTC  

yes you can prove that

2018-10-22 07:30:46 UTC  

it's quite easy actually

2018-10-22 07:31:14 UTC  

the degrees part htat is

2018-10-22 07:31:28 UTC  

we know the exact # of elements in F[x]/f1(x)

2018-10-22 07:31:35 UTC  

it's going to be q^n

2018-10-22 07:31:41 UTC  

i'm assuming q is prime btw

2018-10-22 07:32:18 UTC  

Sure since it's irreducible. Otherwise, hm...

2018-10-22 07:32:58 UTC  

If a polynomial has a non-prime degree, to guarantee reducibility you need to allow complex coefficients, right?

2018-10-22 07:33:48 UTC  

i don't think it matters what the degree of the polynomial is

2018-10-22 07:33:58 UTC  

like if it's prime or not

2018-10-22 07:34:06 UTC  

i think what matters are the coefficients

2018-10-22 07:34:20 UTC  

if you don't have prime coefficients, your polynomials stop being a field

2018-10-22 07:34:42 UTC  

the reason why F[x] is a field is because Z_q is a field when q is prime

2018-10-22 07:34:48 UTC  

if q is not prime, Z_q is not a field

2018-10-22 07:35:01 UTC  

and therefore F[x] is not guaranteed to be a field

2018-10-22 07:35:24 UTC  

Why is Z_q not a field when q is, say, 4?

2018-10-22 07:35:27 UTC  

the very basic example is that say if you have Z_4, the polynomial "2" does not have a multiplicative inverse

2018-10-22 07:35:48 UTC  

Ohh

2018-10-22 07:35:53 UTC  

Neat

2018-10-22 07:35:54 UTC  

Z_4 is not a field because 2 does not have a multiplicative inverse

2018-10-22 07:36:10 UTC  

Yeah haha been a while

2018-10-22 07:36:12 UTC  

and since Z_q is a subfield of F_q[x]

2018-10-22 07:36:26 UTC  

yeah I've literally pullen out my abstract alg. book out today

2018-10-22 07:36:32 UTC  

i need some cryptography knowledge from it

2018-10-22 07:36:43 UTC  

i learned all of this again today 😛

2018-10-22 07:39:45 UTC  

That's neat. I enjoyed abstract algebra but mostly only needed the linear algebra part since then (11 years ago). Rotation groups and some other stuff relevant to geometry, but not much more. Cryptography seems interesting, we did cover RSA at some point, I only remember the main idea, that factoring is hard 😁

2018-10-22 07:41:23 UTC  

@ThisIsChris wellyeah, some pajeets found a way to crack RSA with you guessed it, Galois fields

2018-10-22 07:42:06 UTC  

Is that a recent thing?

2018-10-22 07:42:10 UTC  

yeah

2018-10-22 07:42:12 UTC  

like a year or so

2018-10-22 07:42:43 UTC  

it's amazing how irrelevant Abstract Algebra used to be until computers, and now it's hella important

2018-10-22 07:42:47 UTC  

Huh no kidding. Guess I better put 2FA on my bank accounts then

2018-10-22 07:45:28 UTC  

yep

2018-10-22 07:45:36 UTC  

idk about you, my dad make me take applied math

2018-10-22 07:45:49 UTC  

it was "fun" i guess, but I really want to go back to college and get a pure math degree

2018-10-22 07:45:52 UTC  

later

2018-10-22 07:45:53 UTC  

😛

2018-10-22 07:46:04 UTC  

pure math is so much more fun that learning about how to earn money at google