1,237
edits
Changes
mLine 7:
Line 7: − +
no edit summary
# The [[real numbers]] <math>\mathbb R</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 [[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 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 (mathematics)|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 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>.