Changes

# 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 rational.

# 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 rational.

# 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>.

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

== Axioms ==

== Axioms ==

Technically, a field <math>F</math> is a set of elements endowed with two [[Binary operation|binary operations]], <math>+</math> and <math>\cdot</math> (with properties analogous to addition and multiplication, respectively), which obey the following axioms:

Technically, a field <math>F</math> is a set of elements endowed with two [[binary operation]]s, <math>+</math> and <math>\cdot</math> (with properties analogous to addition and multiplication, respectively), which obey the following axioms:

*Addition axioms

*Addition axioms