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Fields

A setFFFwith two binary operations +,⋅:F×F→F+,\cdot:F\times F\to F+,⋅:F×F→Fis a field if the following holds:

  • R,+R,+R,+is a commutative group with identity 000

  • R−{0},â‹…R-\{0\},\cdotR−{0},â‹…is a commutative group with identity111.

  • Distributivity:

// field extensions, algebraic elements

a(b+c)=ab+ac,(a+b)c=ac+bca(b+c)=ab+ac,(a+b)c=ac+bca(b+c)=ab+ac,(a+b)c=ac+bc