Question

What is group theory, field theory, and ring theory?

Answer 1

george i think you need to use wikipedia :p

Answer 2

or google

Answer 3

http://en.wikipedia.org/wiki/Group_theory

Answer 4

What are the differences between them?

Answer 5

lol idk reading

Answer 6

http://www.quadibloc.com/math/abaint.htm

Answer 7

check that link out

Answer 8

A group is defined as: a set of elements, together with an operation performed on pairs of these elements such that:

The operation, when given two elements of the set as arguments, always returns an element of the set as its result. It is thus fully defined, and closed over the set.
One element of the set is an identity element. Thus, if we call our operation op, there is some element of the set e such that for any other element of the set x, e op x = x op e = x.
Every element of the set has an inverse element. If we take any element of the set p, there is another element q such that p op q = q op p = e.
The operation is associative. For any three elements of the set, (a op b) op c always equals a op (b op c).
A consequence of the third property is that there are no duplicate elements in any row or column of the operation table for a group. A consequence of the fourth property, together with the others, is that every finite group can be expressed as a set of permutations of n objects for some n, where the operation for the group is applying the second permutation to the elements of the first permutation.

There are many different kinds of finite groups, some with very complex structure. Most groups belong to families of groups with an infinite number of members. Thus, addition modulo 5 yields the cyclic group of order 5, and there are cyclic groups of every integer order starting with 2. However, there are 26 groups that don't belong to these infinite families, called sporadic simple groups.

A ring is a set of elements with two operations, one of which is like addition, the other of which is like multiplication, which we will call add and mul. It has the following properties:

The elements of the ring, together with the addition operation, form a group.
Addition is commutative. That is, for any two elements of the set p and q, p add q = q add p. (The word Abelian is also used for "commutative", in honor of the mathematician Niels Henrik Abel.)
The multiplication operation is associative.
Multiplication distributes over addition: that is, for any three elements of the group a, b, and c, a mul ( b add c ) = (a mul b) add (a mul c).
Addition and multiplication modulo 5 and modulo 6 both yield rings. Matrix multiplication also leads to rings as well.

A field is a ring in which the elements, other than the identity element for addition, and the multiplication operator, also form a group.

There are only two kinds of finite fields. One kind is the field formed by addition and multiplication modulo a prime number. The other kind of finite field has a number of elements that is a power of a prime number. The addition operator consists of multiple independent additions modulo that prime. The elements of the field can be thought of as polynomials whose coefficients are numbers modulo that prime. In that case, multiplication is polynomial multiplication, where not only are the coefficients modulo that prime, but the polynomials are modulo a special kind of polynomial, known as a primitive polynomial. All finite fields, but particularly those of this second kind, are known as Galois fields.

Answer 9

lol copypasta