Algebraic field

A field is a com­mu­ta­tive ring \((R, +, \times)\) (hence­forth ab­bre­vi­ated sim­ply as \(R\), with mul­ti­plica­tive iden­tity \(1\) and ad­di­tive iden­tity \(0\)) which ad­di­tion­ally has the prop­erty that ev­ery nonzero el­e­ment has a mul­ti­plica­tive in­verse: for ev­ery \(r \in R\) there is \(x \in R\) such that \(xr = rx = 1\). Con­ven­tion­ally we in­sist that a field must have more than one el­e­ment: equiv­a­lently, \(0 \not = 1\).

Examples

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