Abstract Algebra | Constructing a field of order 4.

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  • เผยแพร่เมื่อ 30 ต.ค. 2024

ความคิดเห็น • 29

  • @davidescobar7726
    @davidescobar7726 4 ปีที่แล้ว +6

    Fantastic channel ... I don't know how I could have missed this type of content on this platform. Many thanks Professor. Greetings

  • @jordancourtemanche562
    @jordancourtemanche562 4 ปีที่แล้ว +3

    Since you're in characteristic 2, another cool way to calculate (α+1)(α+1) is through the Frobenius map.
    I've really been enjoying your content, especially since you go in so many different interesting directions. And you've been on a tear, putting out new videos!

  • @josh34578
    @josh34578 4 ปีที่แล้ว +8

    Also note that the finite field of order 4 is NOT isomorphic to the integers mod 4. That's a somewhat common misconception.

    • @cameronspalding9792
      @cameronspalding9792 4 ปีที่แล้ว +7

      The integers mod 4 isn’t a field because it’s not a division ring

  • @paul21353
    @paul21353 3 ปีที่แล้ว +3

    At 5:58 Z alpha (alpha) is actually Z2 (alpha)

  • @mac2105
    @mac2105 ปีที่แล้ว

    I've rarely ever had less doubt that someone completely understands something...

  • @kevinsuryapranata
    @kevinsuryapranata 4 ปีที่แล้ว +1

    Very nice explanation, I really thanks for this video Sir.

  • @eh0394
    @eh0394 2 ปีที่แล้ว +1

    thank you king

  • @tLearningTree
    @tLearningTree 5 หลายเดือนก่อน +2

    I don't get the part at 3:18 - 3:55 . Can someone explain to me?

    • @jaddaj5881
      @jaddaj5881 หลายเดือนก่อน

      Me neither. I always just thought Z2/P(x) is just notation to mean a+b alpha. I don’t understand, why he calls x +p(x) -> alpha. Or why alpha^2=alpha+1 then implies Z2/P(x) ={a + b alpha| a,b in Z2}.

  • @mark_tilltill6664
    @mark_tilltill6664 2 ปีที่แล้ว

    Take the product and use synthetic division keep the remainder.

  • @عادلبناصر-غ9ن
    @عادلبناصر-غ9ن 2 ปีที่แล้ว

    thank you sir Please I want to solve practical exercises in fields and groups

  • @howardcheung8304
    @howardcheung8304 2 ปีที่แล้ว +1

    Sir can u make some more videos on finite field?

  • @Jim-vr2lx
    @Jim-vr2lx 2 ปีที่แล้ว

    Can someone help explain why AES , in GF(2^8), uses the irreducible polynomial x^8 + x^4 + x^3 + x +1 while the Twofish algorithm, in GF(2^8), uses a different irreducible polynomial x^8 +x^6 +x^3 +x^2 +1 ? Wont both polynomials produce the same values if there is only one finite field of order 2^8? I was told never to mix polynomials -- pick one polynomial and stick with it.

    • @codegeek98
      @codegeek98 11 หลายเดือนก่อน

      They're not "mixing" them. AES defined the field in one (legal) way; Twofish defined them in a different (also legal) way. They're incompatible, but that doesn't matter, because no one is "hotwiring" or "mixing" internal pieces of AES implementations into Twofish implementations.

    • @codegeek98
      @codegeek98 11 หลายเดือนก่อน

      The different versions of GF(2^8) are _isomorphic_ (8:33), which means “the same, if you squint”-you can map between the elements of the different versions in a way that preserves multiplication Map(a)Map(b)=Map(ab) and vice versa-but the different versions of the field look obviously different from each other, if you're just looking at the polynomials.
      If Wikipedia is to be believed, GF(2^2) is the ONLY case where you have just 1 option of irreducible polynomial.

  • @Adityarm.08
    @Adityarm.08 3 ปีที่แล้ว

    Thank you!!

  • @cameronspalding9792
    @cameronspalding9792 4 ปีที่แล้ว

    I heard that every finite field has a size that is prime or a power of a prime
    I could understand fields of prime order: Zp where p is prime, but I couldn’t understand fields with job prime order

  • @julien31415
    @julien31415 3 ปีที่แล้ว

    Nice !

  • @عادلبناصر-غ9ن
    @عادلبناصر-غ9ن 2 ปีที่แล้ว +1

    من فضلك أريد حل تمارين عملية في حقول ومجموعات👍👍👍👍👍👍

    • @ibrahimn628
      @ibrahimn628 2 ปีที่แล้ว

      Any standard abstract algebra textbook have ton of exercises. Try it on your own and seek help from online forums If you’re stuck

  • @darkguild8624
    @darkguild8624 2 ปีที่แล้ว +1

    Why x^2+x^2 become zero?

    • @henk7747
      @henk7747 2 ปีที่แล้ว +2

      Coefficients of polynomials are in Z_2. 1+1=0

  • @vanessamichaels9512
    @vanessamichaels9512 8 หลายเดือนก่อน +1

    why is 2x^2=0?
    why are you screaming?

  • @maxamedaxmedn6380
    @maxamedaxmedn6380 4 ปีที่แล้ว +4

    I think i can't understand this type of math

  • @KurdaHussein
    @KurdaHussein ปีที่แล้ว

    but how x³ , x⁴ are not needed ???

    • @praneeshkumar7341
      @praneeshkumar7341 ปีที่แล้ว +2

      I think you can take any polynomial p that’s degree at least 2 and use Euclidean division to express it as p = (x² + x + 1)q + r. So then modulo x² + x + 1, this just looks like r which is degree strictly less than 2.