Re: (finite)galois field

Hello, thanks to everyone for helping. Now i have a better
understanding. But i have still a lot of things that i cant
understand. I am sorry i coulndt replay since i havent any time last
days.
Galois showed that in general, given any prime p and any positive
integer n, there is one and only one field that can be constructed on
a set with p^n elements; and moreover, that if m is any positive
integer, then there is a field with exactly m elements if and only if
m=p^n for some prime p and positive integer n. These fields are
constructed by starting with F_p, and then "adding" the roots of some
polynomial equation of degree n that is irreducible over F_p. For
example, to get a field of order 3^2=9, you start with F_3, then take
a polynomial equation of degree 2 that is irreducible (x^2+1 = 0
works). Let a be an element such that a^2 + 1 = 0, that is, a
"solution" to this equation. Then the set

{0, 1, 2, a, 2a, 1+a, 1+2a, 2+a, 2+2a}

How did u construct this field? Isnt there any other field? why?
And for what do I use this field?
Can Galois fields be used in solutions of linear equations such
that the unknowns and coefficients are integers where number of
unknowns are less than the equations? Thanks a lot.

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