20171115, 01:22  #1 
"Sam"
Nov 2016
5×67 Posts 
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......q3, q2, q1]. Addition, subtraction, multiplication, and division are performed on these elements. If e is an element in F(q), then e^(q1) = 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^21) = 1.  Show that there are exactly phi(q^21) primitive elements e such that q^21 is the smallest integer m such that e^m = 1. Can someone please provide further information on this? Thank you. 
20171115, 02:17  #2  
"Forget I exist"
Jul 2009
Dumbassville
3·2,797 Posts 
Quote:
as to the second point I'm not sure. 

20171115, 09:42  #3  
Dec 2012
The Netherlands
5×353 Posts 
Quote:
For the second, see theorem 90 and proposition 88. They are all on this page: http://www.mersenneforum.org/showthread.php?t=21904 

20171115, 14:37  #4 
Feb 2017
Nowhere
1011100010111_{2} Posts 
As Nick has already pointed out, only the nonzero elements of a finite field can have a multiplicative order.
Exercise: In any field, a finite multiplicative group is cyclic. Exercise: Let p be a prime number, and let K be a field of characteristic p. Show that, in K, a) (x + y)^p = x^p + y^p for every x, y in K [the "Freshman's dream"] b) If k is the field of p elements, show that x^p = x for every x in k. c) If K is a field of characteristic p, f is a positive integer, and q = p^f, then (x + y)^q = x^q + y^q Exercise: Let k be the field with p elements, f a positive integer, and K the field of q = p^f elements. Determine the number of elements in K having degree f over k. 
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