MathGloss

A field is a set $F$ together with two binary operations addition ($+$) and multiplication ($\cdot$) such that

• Addition and multiplication are both associative;
• Addition and multiplication are both commutative;
• There exist two distinct elements $0\neq 1$ in $F$ such that $0$ is an additive identity and $1$ is a multiplicative identity;
• Every element of $F$ has an additive inverse;
• Every nonzero element of $F$ has a multiplicative inverse;
• Multiplication distributes over addition as follows: $a\cdot (b+c) = (a\cdot b) + (a\cdot c)$ for all $a,b,c\in F$.

Equivalently, a field is a commutative division ring.

Wikidata ID: Q190109

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