Jump to Linear algebra and commutative algebra - In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.‎Examples · ‎Finite fields · ‎Constructing fields · ‎Fields with additional. Fields are very important to the study of linear algebra. This is because any result in linear algebra applies to all fields, because the basic operations in linear algebra involve only addition, subtraction, multiplication, and division. Page 1. I LINEAR ALGEBRA. A. Fields. A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field).


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I A Field in linear algebra addition to familiar number systems such as the rationals, there are other, less immediate examples of fields.

The following example is a field consisting of four elements called O, I, A, and B.

The notation is chosen such that O plays the role of the additive identity element denoted 0 in the axioms aboveand I is the multiplicative identity denoted 1 in the axioms above. The field axioms can be verified by using some more field in linear algebra theory, or by direct computation.

Linear Algebra/Fields

This field is called a finite field with four elements, and is denoted F4 or GF 4. In field in linear algebra context of computer science and Boolean algebraO and I are often denoted respectively by false and true, the addition is then denoted XOR exclusive orand the multiplication is denoted AND.

In other words, the structure of the binary field is the basic structure that allows computing with bits. Elementary notions[ edit ] In this section, F denotes an arbitrary field and a and b are field in linear algebra elements of F.

This means that every field is an integral domain. The additive and the multiplicative group of a field[ edit ] The axioms of a field F imply that it is an abelian group under addition.

Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number.

It is usually denoted by p and the field is said to have characteristic p then. Therefore, the Frobenius map Fr: Subfields and prime fields[ edit ] A subfield E of a field F is a subset of F that is a field with respect to the field operations of F.

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Equivalently E is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. Field homomorphisms are maps f: All field homomorphisms are injective.


A field is called a prime field if it has no proper i. Any field F contains a prime field.

Field (mathematics) - Wikipedia

If the characteristic of F is p a prime numberthe prime field is isomorphic to the finite field Fp introduced below. Otherwise the prime field is isomorphic to Q. Finite field Finite fields also called Galois fields are fields with finitely many elements, whose number is also referred to as the order of the field.

The above introductory example Field in linear algebra is a field with four elements.

The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. The addition and multiplication on this set are done by performing the operation in question in the set Z of integers, dividing by n and taking the remainder as result. This construction yields a field precisely if n is a prime number.

This statement holds since F may be viewed as a vector space over its prime field. The dimension of this vector space is necessarily finite, say field in linear algebra, which implies the asserted statement. Such a splitting field is an extension of Fp in which the polynomial f has q zeros.

This means f has as many zeros as possible since the degree of f is q. By contrast, in F2, f has only two zeros namely 0 and 1so f does not split into linear factors in this smaller field.

Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. History[ edit ] Historically, three algebraic disciplines led to the concept field in linear algebra a field:

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