Finite fields made easy

Solutions to some typical exam questions. See my other videos

[WAMProducties]

Great video! I'm trying to understand this stuff off wikipedia and all I'm doing is frying my brain. This really helped, thanks!

[romywilliamson]

Thank you!!! I have a Galois theory exam in 2 weeks and loads of the exam questions involve finite fields but no one ever taught us what they are. For ages I've been looking for a nice intuitive explanation with concrete examples, this is perfect.

[NassosKranidiotis]

Thank you so much for the excellent video!

[ManuelBTC]

It would have been better to be explicit at 2:50
0x + 0 1x + 0 2x + 0
0x + 1 1x + 1 2x + 1
0x + 2 1x + 2 2x + 2

[PETAJOULE]

Great video explaining field construction for simple cases and also for complex cases which utilize polynomials instead (method2 with alpha is little different than from own source). Also good note that number of elements of finite field must be related to prime or its powers.

[kagayakuangel]

Question, for this contradiction, at 5:37 when you say it is impossible for a^2d^2 = 2 you simply mean that it is impossible to find 2 numbers in the set {0, 1, 2} such that their squares can multiply to 2 right?
like
0^2 • 0^2 = 0 ≠ 2
0^2 • 1^2 = 0 ≠ 2
0^2 • 2^2 = 0 ≠ 2
1^2 • 1^2 = 1 ≠ 2
1^2 • 2^2 = 4 ≡ 1 ≠ 2
(got lazy here not writing the rest of this out)
⋮
and so on...

[vaishnavj]

Should the polynomial we are dividing be irreducible or primitive?

[butttasdtics]

At 5:52 how does 2x^2 + 1 = 2(x^2 + 1) + 2?

I feel like there's a step missing or something I'm just not seeing.

[gilian]

Determining which irreducible polynomial is associated with your generic GF(p^n) is the piece that I don't quite understand. GF(2^8) provides an irreducible polynomial equivalent to the hexadecimal value of 0x11B; I know this, that's what I've read; but I have no clue why, or how to derive that result for myself. Calculating result of multiplying (0x32).(0x62) on the Finite Field by hand seems to be a carpal tunnel inducing exercise; and you need to do perform this multiplication around 320 times for the AES algorithm to encrypt just one 16 Byte string. This whole thing makes me suspect that my brain might be leaking out of my left ear...

[thenapoler]

Thank you for this video, really helped me understand fields better.

[oborooizamisi]

Hello, please I am still confused on how you are doing the modular reduction in g(x) after multiplication?

Then how do we add and multiply:
7 and 8 for example

[rutgers]

Love the video! I'm trying to understand how you come up with the 9 polynomials at 2:52 from F sub 9?

[rutgers]

why did you do mod3 instead of mod9 at 3:59. Is it always the base prime (3 for 27, 5 for 125) for a finite field of an order of a prime power?

[Post]

You say a number must be invertible to be in the set, but is there any case where 0 has an inverse, where 0*x = 1?

[smashingdots]

So at 4:30, how is (2x+1)(x+1)=2x^2 + 1 ???? What happens to the 3x?

[Aitchdeee]

At 2:38, how does the fact that 9 = 3^2 let us produce the 9 polynomials?

[JohnDoe-sczf]

Hey, can you explain why Z2 = {0, 1} with addition and multiplication modulo 2 is a field? Since, I believe 1 is the multiplicative identity, but multiplicative inverse of 0 does not exists i.e 0 * ? = 1. Then how can z2 be a field. Btw awesome video.

[willsharp]

Can alpha^2 + 1 be simplified to 0 because alpha is the imaginary number i?

[thulasiraju]

Sir this is this number theory topic please reply me sir

[phazei]

Really would like a better explanation of 6:15.... 2x^2 = 2x^2 + 0 = 2x^2 + 3 = 2x^2 + 2 + 1 = 2(x^2 + 1) + 1. x^2+1⟌2x^2 + 2 + 1 = 2 + 1/(x^2 + 1) I don't understand the division...