Examples of groups:

A group might be commutative, called 'Abelian', or it might be 'non-Abelian'..
Abelian / Commutative: For all g and h in G, we have gh = hg.

The groups {R, +},{Q, +},{Z, +} are all Abelian because a + b = b + a.

The group M of 2x2 matrices (with non-zero determinant) under matrix mult. is NOT Abelian.

The identity element is I = {{1, 0},{0, 1}}; inverses work out nicely, it's associative, but
the two matrix products AB and BA are very often different (you look for an example)
There are many other types of groups: finite groups (not infinite as sets), p-groups (groups of size p^k),
normal subgroups, symmetry groups of geometric objects, permutation groups, Galois groups.

Rings and Ideals (top of page)
A ring is a set R (not the real numbers R) with two operations, + and * , such that:
R is closed under both ops, R is an abelian group under +, and has distributive laws.
Most rings dealt with at first are commutative rings with identity (for multiplic'n).
Some famous rings: the 'ring of integers' Z, (or rationals Q, reals R, complex nos C) with
usual addition and multip. For example, the set M of 3 by 3 matrices forms a ring under
addition A+B and matrix multiplication AB. Scalar multiplication kA isn't used in this case.
A subset S of a ring R that's also a ring is called a subring ; I write S < R for subring.
A subset J of R is an ideal of R if J is an additive subgroup of R, and if all products j r
(with j in J, r in R), are in J. For example the subring 2Z of even integers is an ideal of Z.

Fields and finite extensions of the rationals (top of page)
Just as non-zero numbers have reciprocals and square matrices (with determ. =/= 0)
have inverses, we define a field as a set K that's a ring as above but with the extra
condition that all nonzero elements x have an inverse x^(-1) ; general notation, not
nec. reciprocals or inverse functions, although those are some examples!
If K is any field, a field extension of K is a field L that contains K.
For example, R is an extension of Q because all rationals are real.
Also the field K = Q(i) = {a + b i , i^2 = -1, a, b in Q} is an extension of Q.
The second case is an example of a finite extension of Q meaning that all numbers
in K can be written as linear combinations of 2 things: 1 and i.
This is the same as the dimension of K as a vector space over Q (see Linear Algebra).
The minimal polynomial of an element b in K over Q is the smallest degree p(x)
such that p(b) = 0. For example the min poly for i over Q is p(x) = x^2 + 1; p(i) = 0.

Galois Theory (top of page)
This is a beautiful connection between finite groups of order n and finite extensions of degree n of fields.
In fact the key is the minimal polynomial f(x) of an element of L over K (see above) and how it factors.
For many (finite) extensions L over K, there is a (finite) group G of all K-automorphisms of L meaning
1-1, onto functions f: L --> L with: f(a+b) = f(a) + f(b), f(ab) = f(a) f(b), all a, b in L, f(k) = k, all k in K.
This group G is called the Galois group of the extension L over K. Not all extensions are 'Galois'.
The notion of 'fixed field': the set of autos of L that fix (leave f(a) = a) fields M larger than K form a
subgrp H of G; subfields M < L and subgroups H < G form a 1-1 correspond'ce under 'fixedfieldof'.
Galois also proved that some polynomials of degree 5 or more were unsolvable by radicals, meaning
their roots are not expressible as a finite series of the 4 basic operations along with exponents & roots.
This boils down to showing the Galois group of the splitting field of the polynomial is a 'solvable group'.
The results of Galois Theory also settled the three classical Greek geometric construction problems: that
1) squaring a circle, 2) doubling a cube, 3) trisecting an angle, are all impossible with ruler & compass.


Higher Math (math major stuff)
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