Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: [[Euclidean diffeomorphism]] Justifications and Intuition:: *[[Justifications and Intuition]]* > [!definition] Definition. ([[field]]) >A **field** is a nonzero [[commutative ring]] (with $1$) in which every nonzero element is a [[unit]] (i.e., a [[commutative ring|commutative]] [[division ring]]). > > Explicitly, a field is a set $F$ equipped with two [[commutative]], [[associative]] [[binary operation]]s $+$ and $\times$, such that >- there is a $+$-Identity $0 \in F$ >- there is a $\times$-Identity $1\in F$ > - each $x\in F$ has a $+$-Inverse, denoted $-x$ >- each $x \in F \backslash \{ 0 \}$ has a $\times$-Inverse, denoted $x^{-1}$ >- for all $a,b,c \in F$ we have $a \times (b+c) = a \times b + a \times c$ > - $0 \neq 1$. > [!intuition] > Fields capture the notion of 'friendly scalar'. > [!equivalence] > - [[division ring iff ideals are {0} and R|commutative ring is a field iff ideals are {0} and R]] > - [[integral domain is a field iff its cyclic modules are torsion-free]] > ^equivalence > [!basicproperties] > - Every [[ring homomorphism]] from a [[field]] to a nonzero ring is [[injection|injective]]. In particular, every [[ring homomorphism]] of [[field|fields]] is [[injection|injective]]. (Verified by hand in Aluffi margin) ^properties