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Old 2017-11-15, 01:22   #1
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Nov 2016

33210 Posts
Post Finite field extensions

When working with finite fields F(q) of order q (q is an odd prime), F(q) has exactly q elements [0, 1, 2, 3......q-3, q-2, q-1]. Addition, subtraction, multiplication, and division are performed on these elements. If e is an element in F(q), then e^(q-1) = 1 by Fermat's Little Theorem.

For the construction of F(q^2), choose a quadratic non residue r mod q. Then let s be a symbol such that s^2 = r in the same sense as i is a symbol such that i^2 = -1.

Elements are of the form e = a*s+b in F(q^2) where a and b are reduced integers mod q. Addition, subtraction, multiplication, and division are defined in F(q^2). Like in F(q), there are exactly q^2 elements in F(q^2).

- Show that for any element e in F(q^2), e^(q^2-1) = 1.
- Show that there are exactly phi(q^2-1) primitive elements e such that q^2-1 is the smallest integer m such that e^m = 1.

Can someone please provide further information on this? Thank you.
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