Definition. A linear transformation is called orthogonal if it is distance preserving; that is, if denotes the distance between points and , then
The set of all orthogonal transformations is a group under composition, called the real orthogonal group.
Definition.: Given a figure in the plane, its symmetry group is the family of all orthogonal transformations for which
The elements of are called symmetries.
The wonderful idea of Galois was to associate to each polynomial a group, nowadays called its Galois group, whose properties reflect the behavior of . Our aim in this section is to set up an analogy between the symmetry group of a polygon and the Galois group of a polynomial.
# Rings, Domains and Fields
Definition. A cummutative ring with is a set equipped with two binary operations, addtion: and multiplication: such that:
is an abelian group under addition.
multiplication is commutative and associative.
there is an element with and
the distributive law holds:
From now on, we will write ring instead of “commutative ring with 1.”
Definition. A ring is a domain (or integral domain) if the product of any two nonzero elements in is itself nonzero.
Theorem: A ring is a domain if and only if it satisifies the cancellation law:
Theorem. is a domain if and only if is prime.
Definition. An element is a unit if there exists with . （乘法可逆）
Definition. A field is a ring in which every nonzero is a unit.
- If is prime, then is a field.
Theorem: For every domain , there is a field containing as a subring. Moreover, every element has a factorization:
Proof: Just like , define:
- Addition: .
- Multiplication: .
We call is 's fraction field. And we denote the ring of polynomials over , and the field of rational functions over , whose elements are of the form .
# Homomorphism and Ideals
Definition. If and are rings, then a function is a ring homomorphism if for all :
A ring homomorphism is an isomorphism if it is a bijection, we writes .
We can derive immediately:
Definition. The kernel of a ring map is:
Definition. An ideal in a ring is a subset containing 0 such that:
An ideal in a ring is a proper ideal if .
An ideal is a sub additive group of the ring.
If , is the ideal generated by , which is called the principal ideal generated by , denoted by .
Theorem. If is a ring homomorphism, then is a proper ideal in . Moreover, is an injection if and only if .
Proof: contains is self-evident, and:
so . and , so .
If is an injection, then for , so . Conversely, if , and exists , then , so , contradicts.
Theorem: Let be a proper ideal in a ring . Then the additive abelian group can be equipped with a multiplication which makes it a ring and which makes the natural map a surjective ring homomorphism:
- Addition: .
- Multiplitcation: .
Theorem: (First Isomorphism Theorem) If is a ring homomorphism with , then there is an isomorphism given by .
Theorem: If is a field, then every ideal in is a principal ideal.
Proof: If , then . Otherwise, let be the polynomial of least degree in , then we prove .
is obvious since . For the other direction, for , we have:
by polynomial modulo, where or . Now , if then we have contradicted having the smallest degree. So .
Definition. A ring is called a principal ideal domain if every ideal in is principal.
Definition. Let be a field. A nonzero polynomial is irreducible over if and there is no factorization in with and .
where means the degree of .
Definition. An ideal in a ring is called a prime ideal if it is a proper ideal and or .
Example: for , then the ideal in is a prime ideal if and only if is prime.
If , then , so or .
Otherwise, if is a factorization, then .
Theorem: If is a field, then a nonzero polynomial is irreducible if and only if is a prime ideal.
Assume is a prime ideal. If is not irreducible, i.e. there is a factorization and . Since every non-zero element in should have degree , so contradicts.
On the other direction, If is irreducible and , then , then or , thus or . And we need to prove is a proper ideal. If , then , so we have , which is impossible.
Theorem: A proper ideal in is a prime ideal if and only if is a domain.
Definition: An ideal in a ring is a maximal ideal if it is a proper ideal and there is no ideal with .
Theorem: A proper ideal in a ring is a maximal ideal if and only if is a field.
Theorem: If is a principal ideal domain, then every nonzero prime ideal is a maximal ideal.
Definition. A polynomial splits over if it is a product of linear factors in . Of course, splits over if and only if contains all the roots of , i.e.:
Theorem: If is a field and is irreducible, then the quotient ring is a field containing (an isomorphism copy of) and a root of .
Where the isomorphism is: . And the root is , .
Since , so . So in , we have a root: .
Notice, is the isomorphism from “numbers” to “a set of polynomials”. And once we have a root of in , it doesn’t mean that there exists a root for in . If and only if there exists such that , , then is root for in .
Example: . Where contains a root for .
Theorem(Kronecker) Let where is a field. There exists a field containing over which splits.
If , then we choose and which is linear.
If , without loss of generality, we write where is irreducible. Let , then there exists a root for in . So in , we have:
So by induction, we can split .
Example: , then we compute the splitting field of over .
We factorize into irreducible ones, .
Compute . Here is a trick, let , given , then if and only if , i.e. .
So in , there exists no polynomials with degree . Because:
So the potential items in are:
And we have:
So . And
So in , we have , which correspond to the in the proof.
where is a root for in and satisfies . splits over .
Definition. A field has character 0 if its prime field is isomorphic to , it has character p if it’s isomorphic to .
Theorem(Galois): For every prime and every positive integer , there exists a field having exactly elements.
Proof: let , Then by Kronecker theorem, there exists a field containing over which splits, let’s construct . Since splits, so it has roots. And we need to prove that it has no repeat roots. We have:
And if in some field, then has no repetitive roots in the field.
Example: Let .
So there are four roots: the field containing elements is . When the case is more complicated since , and the coefficients would be ugly as something.
# Galois Group
Definition. If is a field, then an automorphism of is an isomorphism of with itself. If is a field extension, then an automorphism of fixes pointwise if .
And we define the Galois Group as:
Theorem. If has distinct roots in its splitting field , then is isomorphic to a subgroup of the symmetric group .