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Two ways to see it:

a ring that is commutative
a field where inverses might not exist

Division ring by

Ciro Santilli 40 Updated 2025-07-16

Two ways to see it:

a ring where inverses exist
a field where multiplication is not necessarily commutative

Classification of finite rings by

Ciro Santilli 40 Updated 2025-07-16

M_{n} (F_{q})

accounts for them all, which we know how to do due to the classification of finite fields.

So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)

Field (mathematics) by

Ciro Santilli 40 Updated 2025-07-16

A ring where multiplication is commutative and there is always an inverse.

A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.

And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.

Basically the nicest, least restrictive, 2-operation type of algebra.

Examples:

Multiplicative inverse by

Ciro Santilli 40 Updated 2025-07-16

Distributive property by

Ciro Santilli 40 Updated 2025-07-16

One of the defining properties of algebraic structure with two operations such as ring and field:

a (b + c) = ab + a c

This property shows how the two operations interact.

Classification of finite fields by

Ciro Santilli 40 Updated 2025-07-16

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as

GF (n)

Every element of a finite field satisfies

x^{or d er} = x

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

GF(2) by

Ciro Santilli 40 Updated 2025-07-16

GF(4) by

Ciro Santilli 40 Updated 2025-07-16

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

4 = 2^{2}

, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

0X + 0
0X + 1
1X + 0
1X + 1

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

X^{2} + X + 1

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

(1 X + 0) + (1 X + 1) = (1 + 1) X + (0 + 1) = 0 X + 1

(2)

Multiplication is done modulo

X^{2} + X + 1

, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

(1 X + 0) \times (1 X + 1) = (1 \times 1) X^{2} + (1 \times 1) X + (0 \times 1) X + (0 \times 1) = 1 X^{2} + 1 X + 0

(3)

Without modulo, that would not be one of the elements of the field anymore due to the

1 X^{2}

So we take the modulo, we note that:

1 X^{2} + 1 X + 0 = 1 (X^{2} + X + 1) + (0 X + 1)

(4)

and by the definition of modulo:

(1 X^{2} + 1 X + 0) mod (X^{2} + X + 1) = (0 X + 1)

(5)

which is the final result of the multiplication.

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

Quadratically closed field by

Ciro Santilli 40 Updated 2025-07-16

Vector field by

Ciro Santilli 40 Updated 2025-07-16

A function:

R^{m} \to R^{n}

Algebra over a field by

Ciro Santilli 40 Updated 2025-07-16

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

complex numbers, i.e. $R^{2}$ with complex number multiplication
$R^{3}$ with the cross product
quaternions, i.e. $R^{4}$ with the quaternion multiplication

Division algebra by

Ciro Santilli 40 Updated 2025-07-16

An algebra over a field where division exists.

Evil by

Ciro Santilli 40 Updated 2025-07-16

Things that are not nice such as:

Taboola, Outbrain, and other chumbox
BLOBs
Europe cookie law
adhesive inside mobile phones and more generally, planned obsolescence
Jupyter Notebook
typographical characters that look like ASCII ones, but are not the ASCII ones, e.g. typographical quotes, em-dash. The non-breaking hyphen is not even whitespace, and by def Why not stick to ASCII when ASCII is good enough?
excessive encapsulation
replacement of master and slave terminology from technology
mailing lists. And to add insult to injury, HTML on mailing list messages instead of plaintext.
blank lines in code added by people trying to increase clarity, especially when there is already indentation for that. Every blank line must be preceded by a line comment explaining what the following block is about, or removed.
messaging software that force you to have a mobile phone
advertisements by telephone/SMS
"state" such as global variables and object members, long live functional programming?
mosquitoes, the only intrinsically bad thing about tropical countries
projects with slow compilation times
Microsoft Windows
the 2019 Chinese government
e-learning websites that only allows verified teachers to write content. Cowards who can't handle ranking algorithms.
domain-specific language
a build system without an out-of-tree option
non-linear Git history: stackoverflow.com/questions/20348629/what-are-advantages-of-keeping-linear-history-in-git
visual programming languages like Scratch. Waste of time. Text programming languages are already equally as visual due to indentation:
```
if x == 0:
    x = 1
```
Just make good serious gamedev libraries and integrated development environments for those real languages instead.
software that prevents you from running as root. Let me fucking shoot myself in the foot if I want to. It is better than having to deal with your hand holding bullshit, which is done in a different way for every project. E.g.: stackoverflow.com/questions/17466017/how-to-solve-you-must-not-be-root-to-run-crosstool-ng-when-using-ct-ng/53099177#53099177
Medium
luxury goods
euphemism
closed access academic journals are evil
websites without OAuth
shower room without a window to the exterior (mould!!!)
single programs with their interface split across multiple windows, e.g. GIMP, ZynAddSubFX
graphical user interfaces
logograms
infinitesimals. Just use limit instead, please
country
knowledge olympiads
programming languages without a decent dominating package system
closed source offline software used by millions
exams
security through obscurities
dots in Gmail address
things in websites that look like links, and behave like links, but don't let you middle click to open them on a separate tab
K-pop
numerical computing language
fiscal paradises
when the front-end of an website changes an important permanent state, but the URL does not change
splash screens: you should show boot messages so that people will know what to Google for when things fail. Do you think computer newbies will be afraid and have nightmares?
milk chocolate: why would you eat that instead of dark chocolate if you are older than 10?
to talk about something without giving the real name to not scare off the audience
mathematical symbol that looks like a Greek letter but isn't. Or perhaps mathematical notation in general
when more than two people gather to play a board game or video game, and two or more people start chatting on and on about random subjects rather than concentrating on the game
watching television while eating. Same for reading, or doing basically anything else but eat. The only acceptable activity is talking relaxedly, not about work.
noises coming out of your bicycle. It is so hard to find where they come to fix them!!!
code drop
private cars as opposed to public transport. As a cyclist, you can just see the effect that large roads have on nearby areas, it just destroys nature.
closed standards
double consonants that make no difference to sound. Dilema? Dilemma? Dillema? Dillemma? Please!
social media websites that show stuff from people you don't follow when you don't explicitly want that, including things which are not ads, just random suggestions. Twitter starting being like that cirac a 2022. Facebook got worse around that time. It is a constant fight against those stupid websites.
- Facebook
  - www.quora.com/How-do-we-turn-off-suggested-anything-on-Facebook-so-that-these-never-show-up-again
- Twitter
  - twitter.com/danbarreirokfan/status/1415328096568360964?lang=en-GB
  - www.quora.com/How-do-I-disable-suggested-topics-on-Twitter
socks with short legs that don't protect your ankle/lower calf from cold/scratches/dirt, e.g. liner socks
Presta valves. Why would such a flimsy tech have become so popular compared to the infinitely superior Schrader!

Frobenius theorem (real division algebras) by

Ciro Santilli 40 Updated 2025-07-16

There are 3: real numbers, complex numbers and quaternions.

Notably, the octonions are not associative.

Associative property by

Ciro Santilli 40 Updated 2025-07-16

Algebraic geometry by

Ciro Santilli 40 Updated 2025-07-16

Algebraic curve by

Ciro Santilli 40 Updated 2025-07-16

Elliptic curve by

Ciro Santilli 40 Updated 2025-07-16

An elliptic curve is defined by numbers

a

and

b

. The curve is the set of all points

(x, y)

of the real plane that satisfy the Equation 1. "Definition of the elliptic curves"

y^{2} = x^{3} + a x + b

Equation 1.

Definition of the elliptic curves

Figure 1.
Plots of real elliptic curves for various values of $a$ and $b$
. Source.

Equation 1. "Definition of the elliptic curves" definies elliptic curves over any field, it doesn't have to the real numbers. Notably, the definition also works for finite fields, leading to elliptic curve over a finite fields, which are the ones used in Elliptic-curve Diffie-Hellman cyprotgraphy.

Elliptic curve group by

Ciro Santilli 40 Updated 2025-07-16

The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.

The group operation is called elliptic curve point addition.

Bibliography:

mathoverflow.net/questions/6870/why-is-an-elliptic-curve-a-group