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 
endowed with constants 
and 
(not equal), a unary operation 
and binary operations 
and 
such that:
for all 
in 
for all 
in 
for all 
in 
for all in 
for all in 
for all in 
for all in 
for all in - For all nonzero in , there exists a
in such that 