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= Classifying
{disambiguate=mathematics}
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In <mathematics>, a "classification" means making a list of all possible objects of a given type.

Classification results are some of <Ciro Santilli>'s favorite: <the beauty of mathematics>{full}.

Examples:
* <classification of finite simple groups>
* <classification of regular polytopes>
* <classification of closed surfaces>, and more generalized <generalized Poincaré conjectures>
* <classification of associative real division algebras>
* <classification of finite fields>
* <classification of simple Lie groups>
* classification of the <wallpaper groups> and the <space groups>


Classification

Classification (mathematics)

{tag=Classification (mathematics)}

$M_n(\F_q)$ 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!)


Classification of finite rings

As per <classification of finite fields> those must be of <prime power> order.

<video Finite fields made easy by Randell Heyman (2015)> at https://youtu.be/z9bTzjy4SCg?t=159 shows how for order $9 = 3 \times 3$. Basically, for order $p^n$, we take:
* each element is a polynomial in $GF(p)[x]$, $GF(p)[x]$, the <polynomial over a field>[polynomial ring over the finite field $GF(p)$] with degree smaller than $n$. We've just seen how to construct $GF(p)$ for prime $p$ above, so we're good there.
* addition works element-wise modulo on $GF(p)$
* multiplication is done modulo an <irreducible polynomial> of order $n$
For a worked out example, see: <GF(4)>.


Finite field of non-prime order

{title2=$GL(n, F_m)$}

= $GL(n, m)$
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{title2}

<general linear group> over a <finite field> of order $m$. Remember that due to the <classification of finite fields>, there is one single field for each <prime power> $m$.

Exactly as over the <real numbers>, you just put the finite field elements into a $n \times n$ matrix, and then take the invertible ones.


Finite general linear group

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= Isomorphic
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Something analogous to a <group isomorphism>, but that preserves whatever properties the given algebraic object has. E.g. for a <field (mathematics)>, we also have to preserve multiplication in addition to addition.

Other common examples include isomorphisms of <vector spaces> and <field (mathematics)>. But since both of those two are much simpler than groups in <classification (mathematics)>, as they are both determined by number of elements/dimension alone, see:
* <classification of finite fields>
* <all vector spaces of the same dimension on a given field are isomorphic>
we tend to not talk about isomorphisms so much in those contexts.


Isomorphism

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They come up a lot in many contexts, e.g.:
* <classification of finite fields>


Prime power

{tag=Classification (mathematics)}

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 $x^{order} = x$.

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!


Ciro Santilli @cirosantilli 37

 Incoming links: Classification of finite fields