Finite field extensions

carpetpool · 2017-11-15, 01:22

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.

science_man_88 · 2017-11-15, 02:17

Quote:

Originally Posted by carpetpool

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.

first point follows from the factorization of q^2-1 as (q-1)*(q+1) does it not ??
as to the second point I'm not sure.

Nick · 2017-11-15, 09:42

Quote:

Originally Posted by carpetpool

- 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.

The first is true for all non-zero \(e\) - see Corollary 84 in our Number Theory discussion group.
For the second, see theorem 90 and proposition 88.
They are all on this page:
http://www.mersenneforum.org/showthread.php?t=21904

Dr Sardonicus · 2017-11-15, 14:37

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.

2017-11-15, 01:22	#1
carpetpool "Sam" Nov 2016 2²·83 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......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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2017-11-15, 14:37	#4
Dr Sardonicus Feb 2017 Nowhere 5,791 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.