What is Galois Field

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When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in simple form to understand this ?

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2 Answers

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Galois field is the name that engineers (and especially those studying error correcting codes) use for what mathematicians call a finite field. In applications, the most commonly used Galois field is $\text{GF}(256)$, also called $\text{GF}(2^8)$. Its elements can be thought of as polynomials of degree $7$ or less with binary coefficients ($0$ or $1$). Addition of two field elements is addition of the two polynomials with coefficients being added modulo $2$. Multiplication is polynomial multiplication modulo a polynomial $m(x)$ of degree $8$, that is, multiply the two given polynomials (which may result in a polynomial of degree as much as $14$) and then divide by $m(x)$, throwing away the quotient and keeping only the remainder.

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A Galois field is a finite field (from the Wikipedia article):

In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements.

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