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→Examples: rational, not real
# 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>.
# 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]].
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.
# 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|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>.