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Field**

A field is a ring with commutative and invertible multiplication; The field \((k,+,\times)\) consists of a set \(k\) and binary operations \(+,\times:k\times k\to k\) which are both commutative, associative, and unital (with units \(0_k,1_k\) respectively), such that \(\forall x\in k:\exists (-x)\in k:x+(-x)=0_k\), \(\forall x\in k\backslash\{0_k\}:\exists x^{-1}\in k:x\times x^{-1}=1_k\), and \(\times\) distributes over \(+\), i.e. \(\forall a,b,c\in k:a\times(b+c)=(a\times b)+(a\times c)\). Typically it is also required that \(0_k\neq 1_k\).