We will first define a field.
Definition. A field is a non empty set with two binary operations and such that has commutative unitary ring structure and satisfy the following property:
This means that every element in except for has a multiplicative inverse.
Essentially, a field is a commutative division ring.
1. (rational, real and complex numbers) with standard and operations have field structure. These are examples with infinite cardinality.
2., the integers modulo where is a prime, and and are mod is a family of finite fields.
Fields and Homomorphisms
An embedding is a ring homomorphism from a field to a field . Since the kernel of a homomorphism is an ideal, a field's only ideals are and the field itself, and , we must have the kernel equal to , so that is injective and is isometric to its image under . Thus, the embedding deserves its name.
Definition (Field Extension and Degree of Extension)
- Let F and G be fields. If and there is an embedding from F into G, then G is a field extension of F.
- Let G be an extension of F. Consider G as a vector space over the field F. The dimension of this vector space is the degree of the extension, . If the degree is finite, then is a finite extension of , and is of degree over F.
Examples (of field extensions)
- The real numbers can be extended into the complex numbers
- Similarly, one can add the imaginary number to the field of rational numbers to form the field of Gaussian integers.
Theorem (Existence of Unique embedding from the integers into a field)
Let F be a field, then there exists a unique homomorphism
Proof: Define such that , etc. This provides the relevant homomorphism.
Note: The Kernel of is an ideal of . Hence, it is generated by some integer . Suppose for some then and, since is a field and so also an integral domain, or . This cannot be the case since the kernel is generated by and hence must be prime or equal 0.
Definition (Characteristic of Field)
The characteristic of a field can be defined to be the generator of the kernel of the homomorphism, as described in the note above.
Definition (Algebraic Elements and Algebraic Extension)
- Let be an extension of then is algebraic over if there exists a non-zero polynomial such that
- is an algebraic extension of if is an extension of , such that every element of is algebraic over .
Definition (Minimal Polynomial)
If is algebraic over then the set of polynomials in which have as a root is an ideal of . This is a principle ideal domain and so the ideal is generated by a unique monic non-zero polynomial, . We define the to be the minimal polynomial.
Definition (Splitting Field)
Let be a field, and are roots of . Then a smallest Field Extension of which contains is called a splitting field of over .
Theorem (Existence of Splitting Fields)
Theorem (Order of any finite field)
Let F be a finite field, then for some prime p and .
proof: The field of integers mod is a subfield of where is the characteristic of . Hence we can view as a vector space over . Further this must be a finite dimensional vector space because is finite. Hence any can be expressed as a linear combination of members of with scalers in and any such linear combination is a member of . Hence .
Theorem (every member of F is a root of )
let be a field such that , then every member is a root of the polynomial .
proof: Consider as a the multiplicative group. Then by la grange's theorem . So multiplying by gives , which is true for all , including .
Theorem (roots of are distinct)
Let be a polynomial in a splitting field over then the roots are distinct.