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A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Everyday examples of fields include the [[real numbers]], [[complex numbers]] and the [[rationals]]. The [[characteristic]] of a field must be either 0 or a [[prime number]] ''p''. For each prime number ''p'' and positive integer ''n'', there is a unique (up to [[isomorphism]]) finite field of [[characteristic]] ''p'' whose [[cardinality]] is ''pⁿ''. A field of characteristic 0 is necessarily infinite.

A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Loosely, a field is a collection of entities with well-behaved and compatible addition and multiplication operations.  A few examples serve to illustrate this point.

 

 

# The [[rational numbers]] <math>\mathbb Q</math>, with operations the usual addition and multiplication.

# The [[real numbers]] <math>\mathbb R</math>, with operations the usual addition and multiplication.

# The [[complex numbers]] <math>\mathbb C</math>, with operations the usual addition and multiplication.

# The integers modulo <math>p</math> ( denoted<math>\mathbb Z/p\mathbb Z</math>), where <math>p</math> is prime.  Here the operations are addition and multiplication modulo <math>p</math>.

Observe that if <math>p</math> is not prime, then <math>\mathbb Z/p\mathbb Z</math> is not a field.  For example, the element <math>2 \in \mathbb Z/6 \mathbb Z</math> has no multiplicative inverse modulo 6!  In this case, <math>\mathbb Z/p\mathbb Z</math> has only the structure of a [[ring]].

# The field <math>\mathbb Q[\sqrt{3}]</math> of real numbers of the form <math>a+b\sqrt{3}</math>, where both <math>a</math> and <math>b</math> are real.

# Finite fields: for each prime number <math>p</math> and positive integer <math>n</math>, there is a unique (up to [[isomorphism]]) finite field of [[cardinality]] is <math>p^n</math>.  This field is of characteristic <math>p</math>.

# The set of [[meromorphic function|meromorphic functions]] on a complex [[manifold]], with pointwise addition and multiplication.  For example, the set of meromorphic functions on <math>\mathbb C</math> or the unit disk <math>\Delta \subset \mathbb C</math>.

# The [[p-adic]] fields <math>\mathbb Q_p</math> and <math>\Omega_p</math>, which play a prominent role in [[number theory]].

 

The [[characteristic]] of a field must be either 0 or a [[prime number]] ''p''. A field of characteristic 0 is necessarily infinite.

 

 

[[Category:Algebra]]

[[Category:Algebra]]