Abstract Algebra/Fields

< Abstract Algebra

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.

Examples:

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.

Contents

Fields and HomomorphismsEdit

Definition (embedding)Edit

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.

Field ExtensionsEdit

Definition (Field Extension and Degree of Extension)Edit

  • 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)Edit

  • 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 rationals.

Theorem (Existence of Unique embedding from the integers into a field)Edit

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)Edit

The characteristic of a field can be defined to be the generator of the kernel of the homomorphism, as described in the note above.

Algebraic ExtensionsEdit

Definition (Algebraic Elements and Algebraic Extension)Edit

  • 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)Edit

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.

Splitting FieldsEdit

Definition (Splitting Field)Edit

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)Edit

Finite FieldsEdit

Theorem (Order of any finite field)Edit

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  )Edit

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)Edit

Let   be a polynomial in a splitting field   over   then the roots   are distinct.