[HN Gopher] Learn you Galois fields for great good (2023)
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Learn you Galois fields for great good (2023)
Author : signa11
Score : 129 points
Date : 2025-06-21 00:21 UTC (22 hours ago)
(HTM) web link (xorvoid.com)
(TXT) w3m dump (xorvoid.com)
| gnabgib wrote:
| (2023)
| https://web.archive.org/web/20230815034422/https://xorvoid.c...
| signa11 wrote:
| yes. so ?
| behnamoh wrote:
| of course it's written in Rust! But I was lowkey looking for
| something more Haskell-y, even Lean. And I wish the
| visualizations would continue throughout the chapters.
| defrost wrote:
| If the goal is learning more about Groups, Fields, etc. there
| are several options of what to do alongside reading the text
| here; use the provided rust code, write code of your own in
| language of choice, use pre existing CAS software that has
| abstract algebra operations, use pencil and paper (there were
| not that many CAS options back in the early days of scaling the
| Monster Group .. it was dissected with a mix of envelopes and
| programs).
|
| GAP and MAGMA a worth a look (GAP is included in other math
| software, eg: SAGE and is open source, MAGMA is commercial with
| education discounts and free student options)
|
| * https://en.wikipedia.org/wiki/GAP_(computer_algebra_system)
|
| * https://en.wikipedia.org/wiki/SageMath
|
| * https://magma.maths.usyd.edu.au/magma/
|
| * https://en.wikipedia.org/wiki/Monster_group
| seanhunter wrote:
| Another FOSS option is maxima, although like everything else
| in maxima, support for group theory is a bit weird and isn't
| exactly comprehensive.
| https://maxima.sourceforge.io/docs/manual/maxima_151.html and
| https://maxima.sourceforge.io/docs/manual/maxima_147.html#It.
| ..
|
| And on the paid side, if you have access to it, mathematica
| has group theory support also and a bunch of named groups
| implemented right out of the box including the Monster group
| and the Conway groups https://reference.wolfram.com/language/
| guide/GroupTheory.htm...
| tempodox wrote:
| If you know Mathematica syntax, you could also try Mathics:
|
| https://mathics.org
| pixelpoet wrote:
| The title is a play on https://learnyouahaskell.com so I
| assumed it would be in Haskell, too. (Rust is much more
| accessible to me though.)
| tempodox wrote:
| I love it when the simple stuff is explained in simple language
| that anybody can understand. Like Einstein said:
|
| _Make it simple. As simple as possible. But no simpler!_
| amelius wrote:
| Yeah, sadly some don't seem to understand this (like quanta
| magazine).
| bluepoint wrote:
| You mean they don't understand the "but no simpler" part?
| revskill wrote:
| The problem with algebra teaching is, they just declare a thing
| without explaining the root reason of why it's there in first
| place.
| deepnet wrote:
| Root reason & comp sci application is mentioned near start :
|
| " Many moons back I was self-learning Galois Fields for some
| erasure coding theory applications."
|
| Erasure codes are based on finite fields, e.g. Galois fields.
|
| The author is fraustrated by access to Galois fields for the
| non-mathematician due to Jargon obscucification.
|
| Also large Application section : "
|
| Applications
|
| The applications and algorithms are staggering. You interact
| with implementations of abstract algebra everyday: CRC, AES
| Encryption, Elliptic-Curve Cryptography, Reed-Solomon, Advanced
| Erasure Codes, Data Hashing/Fingerprinting, Zero-Knowledge
| Proofs, etc.
|
| Having a solid-background in Galois Fields and Abstract Algebra
| is a prerequisite for understanding these applications.
|
| "
|
| I sympathise with your fraustration at math articles.
|
| This is not one of them, it is rich and deep. Xorvoid leads us
| into difficult theoretic territority but the clarity of
| exposition is next level - a programmer will grok some of the
| serious math that underpins our field by reading the OP.
| untitled2 wrote:
| Whining about algebra not being in most CS curriculums is just a
| lie. Every university in the world has (if it doesn't, it's not a
| university) maths as a minor regardless of what your major is.
| And everyone I know, including me, took algebra as a minor being
| a CS major (if you didn't, question your choice of career).
| dunefox wrote:
| > Every university in the world has (if it doesn't, it's not a
| university) maths as a minor regardless of what your major is.
|
| That's just not true.
| chrisdew wrote:
| UK Universities don't have majors and minors as the US does.
| __rito__ wrote:
| Wow, wonderful stuff. Thanks for posting!
| JackFr wrote:
| 1) the properties of a field is missing closure under the
| operation. This is kind of assumed from context, but I would
| include it.
|
| 2) the reduction step up multiplication of nth order polynomials
| (to keep them nth order) is missing (or at least I missed it
| after a couple of readings.)
|
| Apart from those quibbles, this was really good overall though. I
| enjoyed it.
| susam wrote:
| A binary operation on a set is closed on the set by definition.
| If an operation isn't closed, then it isn't considered a binary
| operation on that set. Of course, it doesn't hurt to state the
| closure property explicitly.
|
| I have talked a bit more about it in a totally unrelated blog
| post here: https://susam.net/product-of-additive-
| inverses.html#closure-...
| bananaflag wrote:
| Indeed, I was quite pleasantly surprised when the webpage did
| not mention this infamous and ubiquitous "closure".
| m3kw9 wrote:
| Someone should check grammar before posting a title
| c54 wrote:
| It's a riff on the classic resource for learning Haskell:
| https://learnyouahaskell.com/
| graycat wrote:
| For _abstract algebra,_ there is the polished
|
| I.\ N.\ Herstein, {\it Topics in Algebra,\/}
|
| (markup for TeX word processing).
|
| For Galois theory, took an oral exam on what was in Herstein.
|
| For linear algebra where the field is any of the rationals,
| reals, complex, and finite fields there is
|
| Evar D.\ Nering, {\it Linear Algebra and Matrix Theory,\/} John
| Wiley and Sons, New York, 1964.\ \
|
| As I recall, Nering was an Artin student at Princeton.
|
| Some of the proofs for the rational, real, or complex fields
| don't work for finite fields so for those need special proofs.
|
| Had a course in error correcting codes -- it was applied linear
| algebra where the fields were finite.
|
| Linear algebra is usually about finite dimensional vector spaces
| with an _inner product_ (some engineers say _dot product_ ), but
| the main ideas generalize to infinite dimensions and Hilbert and
| Banach spaces.
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