A Ring can be seen as a generalization of a field where:
- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that is not a ring because you can by addition reach the zero matrix.
- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't
Two ways to see it:
- a ring where inverses exist
- a field where multiplication is not necessarily commutative
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!)
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.
Examples:
One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.
A convenient notation for the elements of of prime order is to use integers, e.g. for we could write:which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:
For non-prime order, we see that modular arithmetic does not work because the divisors have no inverse. E.g. at order 6, 2 and 3 have no inverse, e.g. for 2:we see that things wrap around perfecly, and 1 is never reached.
Finite fields made easy by Randell Heyman (2015)
Source. Good introduction with examplesThere's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as .
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!
Video "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order . Basically, for order , we take:For a worked out example, see: GF(4).
- each element is a polynomial in , , the polynomial ring over the finite field with degree smaller than . We've just seen how to construct for prime above, so we're good there.
- addition works element-wise modulo on
- multiplication is done modulo an irreducible polynomial of order
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.
, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):
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.
Without modulo, that would not be one of the elements of the field anymore due to the !
So we take the modulo, we note that:and by the definition of modulo: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.
Note that the vector product does not have to be neither associative nor commutative.
Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples
- complex numbers, i.e. with complex number multiplication
- with the cross product
- quaternions, i.e. with the quaternion multiplication
An algebra over a field where division exists.
Notably, the octonions are not associative.