# Article

# Article

- Mathematics
- Algebra and number theory
- Field theory (mathematics)

# Field theory (mathematics)

Article By:

**Lewis, D. J. **Department of Mathematics, University of Michigan, Ann Arbor, Michigan.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.256300

- Subfields and extensions
- Solutions of polynomial equations

- Valuations
- Automorphisms of fields

- Links to Primary Literature
- Additional Readings

**In algebra, the term field is used to designate an algebraic system or structure containing at least two elements and having two binary rules of composition: addition and multiplication (that is, if a and b are any two elements of the field, then a + b and ab are defined and are elements of the field).** The structure rules are as follows: The elements form an abelian (commutative) group under addition with the additive identity denoted by 0; that is,

*a*+ 0 =

*a*for all elements

*a*. The set of nonzero elements (and there are some since the field has at least two elements) form an abelian group under multiplication with the multiplicative identity denoted by 1. It follows that all nonzero elements have a multiplicative inverse or reciprocal. The two rules of composition are related by the distributive law: (

*a*+

*b*)

*c*=

*ac*+

*bc*for all elements

*a, b, c*. It follows from the distributive law that

*a*· 0 = 0 for all elements

*a*, since 1 ·

*a*= (1 + 0)

*a*= 1 ·

*a*+ 0 ·

*a*, whence 0 = 0 ·

*a*.

*See also:*

**Group theory**

The content above is only an excerpt.

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