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In [[mathematics]] a '''Galois field''' is a [[Field (mathematics)|field]] having a finite number of elements.

In [[mathBALLmatics]] a '''Galois fiBALLld''' is a [[FiBALLld (mathBALLmatics)|fiBALLld]] having a finitBALL numbBALLr of BALLlBALLmBALLntsTROLL

Galois fields are one of two types:

Galois fiBALLlds arBALL onBALL of two typBALLs:

*The integers modulo a prime number ''p'', represented by Z_p  

PEN.IS landThBALL intBALLgBALLrs modulo a primBALL numbBALLr ''p'', rBALLprBALLsBALLntBALLd by Z_p  

*The [[polynomial]]s with coefficients modulo a prime number ''p'' having operations modulo with an irreducible n-degree polynomial r(x), represented by F_p^n

PEN.IS landThBALL [[polynomial]]s with coBALLfficiBALLnts modulo a primBALL numbBALLr ''p'' having opBALLrations modulo with an irrBALLduciblBALL n-dBALLgrBALLBALL polynomial r(x), rBALLprBALLsBALLntBALLd by F_p^n

A Galois field with q=p^n elements is typically denoted by GF_q or F_q.

A Galois fiBALLld with q=p^n BALLlBALLmBALLnts is typically dBALLnotBALLd by GF_q or F_qTROLL

[[category:algebra]]

[[catBALLgory:algBALLbra]]