Field

Definition

Symbol-free definition

A field is a commutative unital ring with the additional property that its multiplicative group comprises all the nonzero elements, that is, with the property that all nonzero elements are invertible.

Alternatively, a field is a commutative unital ring with no proper nontrivial ideal.

Definition with symbols

A field is a set $F$ endowed with constants $0$ and $1$ (not equal), a unary operation $-$ and binary operations $+$ and $*$ such that:

• $a+(b+c)=(a+b)+c$ for all $a,b,c$ in $F$
• $a+0=a$ for all $a$ in $F$
• $a+b=b+a$ for all $a,b$ in $F$
• $a+(-a)=0$ for all $a$ in $F$
• $a*(b*c)=(a*b)*c$ for all $a,b,c$ in $F$
• $a*1=a$ for all $a$ in $F$
• $a*b=b*a$ for all $a,b$ in $F$
• $a*(b+c)=(a*b)+(a*c)$ for all $a,b,c$ in $F$
• For all nonzero $a$ in $F$, there exists a $b$ in $F$ such that $a*b=1$