Difference between revisions of "Field"

Revision as of 23:54, 30 December 2021

A field is a structure in abstract algebra, similar to a group or a ring. Informally, fields are the general structure in which the usual laws of arithmetic governing the operations $+, -, \times$ and $\div$ hold. In particular, the rational numbers $\mathbb{Q}$ , the real numbers $\mathbb{R}$ , and the complex numbers $\mathbb{C}$ are all fields, although there are many others, including subfields of those fields.

Formally, a field $k$ (here the letter $k$ stands for Körper, the German word for a mathematical field) is a set of elements with two operations, usually called multiplication and addition (denoted $\cdot$ and $+$ , respectively) which have the following properties:

$(k,+)$ is an Abelian group with an identity of $0\in k$ .
$(k\backslash\{0\},\cdot)$ (also denoted as $k^{\times}$ ) is also an Abelian group with an identity of $1\in k$ .
Multiplication ( $\cdot$ ) distributes over addition ( $+$ ); for any $a,b,c\in k$ ,

$a\cdot (b+c)=a\cdot b+a\cdot c.$

There is also a unique name for $(k\backslash\{0\},\cdot)$ , which most accept as the group of units of $k$ . The reason why the group of units of $k$ does not contain $0$ is because it would not be a group, since a group must have a single identity. Furthermore, it can be proven that the group of units of $k$ is a cyclic group for any field $k$ which can help in determining certain homomorphisms between fields.

Common examples of fields are the rational numbers $\mathbb{Q}$ , the real numbers $\mathbb{R}$ , or $\mathbb{Z}/p\mathbb{Z}$ (the integers modulo $p$ for some prime $p$ ). In general, a field of order $N$ is denoted as $\mathbb{F}_N$ , although this is rather unspecific since fields are usually referenced by name.

The study of fields and all of their properties is called field theory, where very interesting theorems can be proved such as the Fundamental Theorem of Algebra, the Abel-Ruffini Theorem, and more.

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-A '''field''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[ring]].  Informally, fields are the general structure in which the usual laws of [[arithmetic]] governing the operations <math>+, -, \times</math> and <math>\div</math> hold.  In particular, the [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, and the [[complex number]]s <math>\mathbb{C}</math> are all fields.
+A '''field''' is a structure in [[abstract algebra]], similar to a [[group]] or a [[ring]].  Informally, fields are the general structure in which the usual laws of [[arithmetic]] governing the operations <math>+, -, \times</math> and <math>\div</math> hold.  In particular, the [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, and the [[complex number]]s <math>\mathbb{C}</math> are all fields, although there are many others, including subfields of those fields.
-Formally, a field <math>F</math> is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition (denoted <math>\cdot</math> and  <math>+</math>, respectively) which have the following properties:
+Formally, a field <math>k</math> (here the letter <math>k</math> stands for Körper, the German word for a mathematical field) is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition (denoted <math>\cdot</math> and  <math>+</math>, respectively) which have the following properties:
-* A field is a ring.  Thus, a field obeys all of the ring axioms.
+* <math>(k,+)</math> is an Abelian group with an identity of <math>0\in k</math>.
-* <math>1 \neq 0</math>, where 1 is the multiplicative [[identity]] and 0 is the additive indentity.  Thus fields have at least 2 elements.
+* <math>(k\backslash\{0\},\cdot)</math> (also denoted as <math>k^{\times}</math>) is also an Abelian group with an identity of <math>1\in k</math>.
-* If we exclude 0, the remaining elements form an [[abelian group]] under multiplication.  In particular, multiplicative [[inverse with respect to an operation | inverses]] exist for every element other than 0.
+* Multiplication (<math>\cdot</math>) distributes over addition (<math>+</math>); for any <math>a,b,c\in k</math>,
+<cmath>a\cdot (b+c)=a\cdot b+a\cdot c.</cmath>
+There is also a unique name for <math>(k\backslash\{0\},\cdot)</math>, which most accept as the '''group of units''' of <math>k</math>.  The reason why the group of units of <math>k</math> does not contain <math>0</math> is because it would not be a group, since a group must have a single identity.  Furthermore, it can be proven that the group of units of <math>k</math> is a cyclic group for any field <math>k</math> which can help in determining certain homomorphisms between fields.
+Common examples of fields are the [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, or <math>\mathbb{Z}/p\mathbb{Z}</math> (the [[integers]] modulo <math>p</math> for some prime <math>p</math>).  In general, a field of order <math>N</math> is denoted as <math>\mathbb{F}_N</math>, although this is rather unspecific since fields are usually referenced by name.
+The study of fields and all of their properties is called [[field theory]], where very interesting theorems can be proved such as the [[Fundamental Theorem of Algebra]], the [[Abel-Ruffini Theorem]], and more.
-Common examples of fields are the [[rational number]]s <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, or the [[integer]]s <math>\mathbb{Z}</math> taken [[modulo]] some [[prime]] <math>p</math>, denoted <math>\mathbb{F}_{p}</math> or <math>\mathbb{Z}/p\mathbb{Z}</math>.  In each case, addition and multiplication are defined "as usual."  Other examples include the set of [[algebraic number]]s and [[finite field]]s of order <math>p^{k}</math> for <math>k</math> an arbitrary positive integer.
 [[Category:Field theory]]

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Difference between revisions of "Field"

Revision as of 23:54, 30 December 2021