Field (mathematics)

A body in the mathematical field of algebra excellent algebraic structure, in which the addition, subtraction, multiplication and division in a particular manner can be performed.

The term body was introduced in the 19th century by Richard Dedekind.

The main body, which are used in almost all areas of mathematics, the field of real numbers, the field of rational numbers and the field of complex numbers.

Formal definition

General definition

A body is a lot equipped with two double-digit shortcuts " " and "" ( addition and multiplication are called ) are met for the following conditions:

Itemization of the required axioms

A body must therefore satisfy the following individual axioms:

Definition as a special ring

A commutative unitary ring which is not the zero- ring, is a body, when in it, each nonzero element has an inverse relation to the multiplication.

In other words, a field is a commutative unitary ring, in which the unit group, ie the maximum size is equal.

Comments

The definition ensures that in a body in the " usual " way, addition, subtraction and multiplication work (and the division with the exception of the Division not defined by 0):

• The inverse of with respect to addition, and is usually the additive inverse of, or the negatives of being called.
• The inverse of with respect to multiplication is and will be the ( multiplicative) inverse of or the inverse of called.
• Is the only member of the body, which does not have the inverse of, the multiplicative group of a body is so.

Note: The formation of the negative of an element has nothing to do with the question of whether the element is itself negatively; For example, the negative of the real number is the positive number. In a general body there is no concept of negative and positive elements. ( See also ordered field. )

Generalizations: skew fields and coordinates body

→ Main article: skew field, Ternärkörper

When you remove the condition that the multiplication is commutative, we come to the structure of the skew field. However, there are also authors who explicitly assume a skew field that multiplication is not commutative. In this case a body is not at the same time skew fields. One example is the skew field of quaternions, which is not a body. On the other hand, there are authors, so Bourbaki, the skew field called the body and the body as discussed here commutative body.

In analytic geometry bodies are used for coordinate representation of points in affine and projective spaces, see affine coordinates, projective coordinate system. In the synthetic geometry in which also spaces ( in particular planes) are investigated with weaker properties, to use as a coordinate ranges ( " coordinates body " ) also generalizations of skew fields, namely Alternative Body, body and quasi Ternärkörper.

Properties and concepts

• There is exactly one "0 " (zero - element neutral element with respect to the body - addition) and a " 1" (one element, neutral element with respect to the body - multiplication) in a body.
• Each body is a ring. The properties of the multiplicative group cancel out the body from the rings. If the commutativity of the multiplicative group is not required, one obtains the notion of skew field.
• Every body is zero divisors: A product of two elements of the body is exactly 0 if at least one of the factors is 0.
• Each body can be assigned to a characteristic which is either 0 or a prime number.
• The smallest subset of a body that still satisfies all axioms body itself, its prime field. The prime field is either isomorphic to the field of rational numbers (for fields of characteristic 0) or a finite residue field (for fields of characteristic, especially for all finite fields, see below).
• A body is a one-dimensional vector space over itself as the underlying Skalarkörper. In addition, there are over all bodies vector spaces of arbitrary dimension. (→ Main article vector space ).
• An important means to examine a body is the polynomial algebra of polynomials in one variable with coefficients.

• It's called a body algebraically closed if every nonconstant polynomial can be decomposed into linear factors of from.
• It's called a body completely when no irreducible non-constant polynomial of in any field extension has multiple zeros. Algebraic closedness implies perfection, but not vice versa.

• If in a body, a total order is defined, which is compatible with the addition and multiplication, one speaks of a parent body and is called the total ordering and arrangement of the body. In such bodies can speak of negative and positive numbers.

• If each body element can be surpassed by a finite sum of one element in this array ( ), it is said, the body satisfies the Archimedean axiom or is also Archimedean ordered.

• In valuation theory certain body are investigated using an evaluation function. They are then called evaluated body.
• A body has a ring only the trivial ideals.
• Each non-constant homomorphism of a body in a ring is injective.

Field extension

→ Main article: field extension

A subset of a body, which again forms a body with its own operations, is called sub ​​- or partial-body. The couple and is called field extension, or. For example, the field of rational numbers is a subfield of the real numbers.

A portion of a body is a part of the body when it has the following properties:

• ,
• ( Seclusion with respect to addition and multiplication )
• ( For each element is also the additive inverse in. )
• ( For each element, excluding the zero is also the multiplicative inverse in. )

The algebraic part that deals with the study of body extensions, the Galois theory.

Examples

• Known examples of bodies are

• The set of rational numbers,
• The set of real numbers and
• The set of complex numbers each having the ordinary addition and multiplication.

• Other examples are provided by the residue field and

• Their finite field extensions, finite body
• General whose algebraic field extensions, the Frobeniuskörper, and
• More generally, any of their body extensions, body with prime characteristic.

• For every prime number of the body of the p- adic numbers.

• The set of integers with the usual connectives is not a body: While a group with neutral element, and each has the additive inverse, but is not a group. After all, the neutral element, but except to and there is no multiplicative inverse (for example, is not a whole, but a real rational number ):

• The integers represent only integral domain whose quotient field, the rational numbers.

• The concept, which dilate the integrity of the ring of integers of the field of rational numbers, and can be embedded in these can be generalized to arbitrary integrity Rings:

• This results in the theory of functions of the Integrity ring of holomorphic on a region of the complex plane functions of the body, the meromorphic functions in the same field and abstract
• From the integrity of the ring of formal power series over a field whose quotient field, anlalog from the integral domain of formal Dirichlet series
• Out of the ring of polynomials in variables whose quotient field, the field of rational functions in as many variables.

Finite Fields

→ Main article: finite body

A body is a finite field, if its ground set is finite. The finite elements are fully classified in the following sense: Every finite field has exactly elements with a prime and a positive natural number. Up to isomorphism, for every such exactly one finite field, which is denoted by. Every body has the characteristic. As an example, the addition and multiplication tables are shown the; color highlighted its lower body.

In the special case we get the body that is isomorphic to the residue field at any prime.

History

Key results of the field theory are due to Évariste Galois and Ernst Steinitz.