Finite Fields

(Fraleigh 33.1) Let be a finite extension of degree over a finite field . If has elements, then has elements
- (Fraleigh e29.30) Let where is a finite field with . Let be algebraic over such that . Then
  - Intuition: If is algebraic with degree , then there exists a basis of elements. Each basis must consist of any of the elements of .
(Fraleigh 33.2) If is a finite field of characteristic then contains exactly elements for some .
(Fraleigh 33.3) Let be a field of elements contained in an algebraic closure of . The elements of are precisely the zeros in of the polynomial in .
- The nonzero elements of a finite field of elements are all -th roots of unity
(Fraleigh 23.6) If is a finite subgroup of the multiplicative group then is cyclic. (Fraleigh 33.5) In particular, the group from non-zero elements of a finite field, is cyclic.
- Proof: The multiplicative group is Abelian therefore it can be factored by the Fundamental Theorem of Finitely Generated Abelian Groups . Let . Now for since divides . Thus, every element of is a zero of . But also, since there are at most zeros. Therefore so it is cyclic.
(Fraleigh 33.6) A finite extension of a finite field is a simple extension of .
- Intuition Any that generates the cyclic group of nonzero elements of implies .

Galois Fields

(Fraleigh 33.8) If is a field of prime characteristic with algebraic closure , then has distinct zeros in .
- Intuition factors into linear factors. Let be the root of one such linear factor. It can be shown that each summand in is in . Since the field has prime characteristic, , it can be shown that . Thus has multiplicity and there are such .
(Fraleigh 33.9) If is a field of prime characteristic , then and .
(Fraleigh 33.10) The Galois Field of order denoted of elements exists for every prime power.
- Intuition: It is the set of all zeros of in the algebraic closure .
(Fraleigh 33.11) If is any finite field, then for every , there is an irreducible polynomial in of degree .
(Fraleigh 33.12) Let be prime and . If and are fields of order then .