Difference between revisions of "Field (mathematics)"
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# 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>. | ||
Revision as of 01:23, January 12, 2014
A field is a commutative 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.
Examples
- The rational numbers
, with operations the usual addition and multiplication. - The real numbers
, with operations the usual addition and multiplication. - The complex numbers
, with operations the usual addition and multiplication. - The integers modulo
(denoted 
), where is prime. Here the operations are addition and multiplication modulo . Observe that if is not prime, then is not a field. For example, the element 
has no multiplicative inverse modulo 6! In this case, has only the structure of a ring. - The field
of real numbers of the form 
, where both 
and 
are rational. - Finite fields: for each prime number and positive integer
, there is a unique (up to isomorphism) finite field of cardinality is 
. This field is of characteristic . - The set of meromorphic functions on a complex manifold, with pointwise addition and multiplication. For example, the set of meromorphic functions on or the unit disk
. - The p-adic fields
and 
, 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.
Fields play an important role in nearly every area of mathematics, and are one of the most basic objects studied by algebra. The study of the relationships between different fields, and in particular subfields of a given field, leads to the study of Galois theory, and makes possible the proof of Abel's theorem and was one of the motivations for the early study of fields and abstract algebra more generally.
Axioms
Technically, a field F consists of a set of elements and two binary operations, addition + and multiplication *, which obey the following axioms:
Addition
1. Closure: If a,b ∈ F then a + b ∈ F.
2. Associativity: For a,b,c ∈ F, (a + b) + c = a + (b + c).
3. Commutativity: For a,b ∈ F, a + b = b + a
4. Identity: There exists an element 0 ∈ F such that a + 0 = 0 + a = a for all a ∈ F.
5. Inverse: If a ∈ F then there exists an (-a) ∈ F such that a + (-a) = (-a) + a = 0.
Multiplication
1. Closure: If a,b ∈ F then a*b ∈ F.
2. Associativity: For a,b,c ∈ F. (a*b)*c = a*(b*c).
3. Commutativity: For a,b ∈ F, a*b = b*a
4. Identity: There exists an element 1 ∈ F such that a*1 = 1*a = a for all a ∈ F.
5. Inverse: If a ∈ F and a ≠ 0 then there exists an a-1 ∈ F such that a*a-1 = a-1*a = 1.
Distributivity
For a,b,c ∈ F, a*(b + c) = a*b + a*c.
Sometimes the condition 0 ≠ 1 is also included.
