field

Definition

Define a field $F$ as a set together with two laws of composition,

• $F \times F \stackrel{+}\to F$, addition: $a,b \leadsto a + b$
• $F \times F \stackrel{\times}\to F$, multiplication: $a, b \leadsto ab$
  such that the axioms below are satisfied,
• addition makes $F$ into an abelian group $F^+$ with identity element denoted $0$
• multiplication is commutative and makes the set of nonzero elements of $F$ into an abelian group $F^\times$, with its identity element denoted $1$
• distributive law, for all $a,b,c \in F$, $a (b+c) = ab + ac$

Notes

• not to be confused with σ-algebra from analysis (which is also known as a "field")

See also

• ring
• group
• subfield

References

1. M. Artin, Algebra, 2. ed. Boston, Mass. Munich: Pearson Education, Prentice Hall, 2011, pp. 80-81.
2. https://www.ccs.neu.edu/home/wichs/class/crypto-fall17/lecture2.pdf
3. https://ncatlab.org/nlab/show/field
4. https://www.bananaspace.org/wiki/域
5. https://math.stackexchange.com/questions/75/what-are-the-differences-between-rings-groups-and-fields
6. https://mathworld.wolfram.com/Field.html