Abstract Algebra/Fields

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Definition: A field is a non empty set F with two binary operations + and such that (F, + ,) has commutative unitary ring structure and satisfy the following property:

(every element in F except for 0 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 fields examples with infinite cardinality.

2.- , the integer set modulo p (p a prime positive integer number) with and operations is a family of finite fields.

Contents 1 Exercises 2 Field Extensions 2.1 Definitions 3 Fields and Homomorphisms 4 Theorem 4.1 Proof

[edit] Exercises

1. Prove that the above examples are fields.

[edit] Field Extensions

Consider the polynomial X² + 1 over the field of real numbers. This polynomial has no roots. However, if we consider the polynomial over the larger field of the complex numbers, then we get the two roots i and -i. Thus, when we consider a polynomial over a larger field, we can get new roots. This is called an extension.

[edit] Definitions

• Let F and G be fields, and let . Then G is a field extension of F.
• Let G be an extension of F. Now consider G as a vector space over the field F. The dimension of this vector space is the degree of the extension, (G:F). If the degree , then G is a finite extension of F, and that G is of degree n=(G:F) over F.

[edit] Fields and Homomorphisms

Suppose we have a homomorphism from a field F to a field G. Then it either maps all of its elements to 1_G, or it is injective or one-to-one. This is because its only proper ideal is trivial.

[edit] Theorem

Suppose that f is a non-constant polynomial within F[X]. Then there exists an extension of F which contains an element that is a root of f.

[edit] Proof