1. ## finite field homework

It's just about halfway through the term and I've been able to do the first 5 homeworks without any help, however my class notes seem to be lacking substance at this point.
So I come to you for help.

2. Show that $\displaystyle X^3-X+1$ is irreducible in $\displaystyle \mathbb{F}_3[X]$
if $\displaystyle L=\mathbb{F}_3[X]/(X^3-X+1)$

my solution so far: $\displaystyle (x), (x+1), (x+2), (x2), (2x+1), (2x+2)$ aren't roots so it's irreducible
then from an example we did in class I need to find the number of elements in this field, then the number of elements in the cyclic group of nonzero elements? then what we did next in the example is find an element of this group of degree equal to the size of the group?

what I'd like to improve on this, I don't particularly like my proof of irreducibility since I have another question which is of degree 4 so I don't want to have to list all examples again, but it is $\displaystyle \mathbb{F}_2$ so it's still not hard to do it exhaustively

3. Let $\displaystyle K$ be a finite field of characteristic $\displaystyle p$, $\displaystyle F$ a subfield of $\displaystyle K$, and $\displaystyle m$ a positive integer.
set $\displaystyle L= \{ a \in K:a^{p^{m}} \in F \}$

a) show that $\displaystyle L$ is a subfield of $\displaystyle K$ that contains $\displaystyle F$
b) show that $\displaystyle L=F$

5. Let $\displaystyle p$ be a prime and $\displaystyle f(x)$ an irreducible polynomial of degree 2 in $\displaystyle \mathbb{F}_p[X]$. if $\displaystyle K$ is an extension field of $\displaystyle \mathbb{F}_p$ of order $\displaystyle p^3$, prove that $\displaystyle f(x)$ is irreducible in $\displaystyle K[X]$

This is due in about 12 hours from now, thanks for the help. I'm sorry I left it to the last minute, I tried hard to do it on my own
Even if you only get to this after the deadline I'd still appreciate someone guiding me through this as I want to know this stuff for my midterm next week too

2. Originally Posted by jbpellerin
It's just about halfway through the term and I've been able to do the first 5 homeworks without any help, however my class notes seem to be lacking substance at this point.
So I come to you for help.

2. Show that $\displaystyle X^3-X+1$ is irreducible in $\displaystyle \mathbb{F}_3[X]$
if $\displaystyle L=\mathbb{F}_3[X]/(X^3-X+1)$

== This question's odd: the pol. f(x) = x^3 - x + 1 is irreducible over Z_3 no matter what. I don't understand what the meaning of the line beginning with "if" in your question can possibly be.

Anyway, since the pol. f(x) is irreduc. the ideal generated by it is prime and thus maximal (since the pol's ring is a PID) and thus the quotient ring is a a field.

my solution so far: $\displaystyle (x), (x+1), (x+2), (x2), (2x+1), (2x+2)$ aren't roots so it's irreducible

== This again is weird: x, x+ 1 and etc. are NOT candidates to be roots of
f(x) since these are pol's, NOT elements of the field F_3.
In fact, f(x) is irred. because its degree is <= 3 and it has no roots in the definition field, namely F_3

then from an example we did in class I need to find the number of elements in this field, then the number of elements in the cyclic group of nonzero elements?

What's then question here? The number of elements in F_3[x]/<f(x)> is 3^deg(f(x)) ==> 3^3 = 27.
As L is a field, L* is a multiplicative group, and since it is a finite one then it is cyclic.

then what we did next in the example is find an element of this group of degree equal to the size of the group?

Are you asking people what YOU did in class? Anyway, perhaps you mean to find an element w in L s.t. L* = <w>. Obviously it must be an element of order 26.

what I'd like to improve on this, I don't particularly like my proof of irreducibility since I have another question which is of degree 4 so I don't want to have to list all examples again, but it is $\displaystyle \mathbb{F}_2$ so it's still not hard to do it exhaustively

To show irreducibility for pol's of degree > 3 can be way harder since one cannot use the simple test of non-existence of root as with pol's of degree <= 3.

3. Let $\displaystyle K$ be a finite field of characteristic $\displaystyle p$, $\displaystyle F$ a subfield of $\displaystyle K$, and $\displaystyle m$ a positive integer.
set $\displaystyle L= \{ a \in K:a^{p^{m}} \in F \}$

a) show that $\displaystyle L$ is a subfield of $\displaystyle K$ that contains $\displaystyle F$
b) show that $\displaystyle L=F$

=== Do you already know that any finite field K of order p^n, with p a prime and n a natural number, is in fact the splitting field of the pol. x^(p^n) - x over the prime field F_p? This means that K is in fact the set of all the roots of the above pol. (in some algebraci closure of F_p) and it in fact is a field wrt the standard addtion and multiplication operations.

5. Let $\displaystyle p$ be a prime and $\displaystyle f(x)$ an irreducible polynomial of degree 2 in $\displaystyle \mathbb{F}_p[X]$. if $\displaystyle K$ is an extension field of $\displaystyle \mathbb{F}_p$ of order $\displaystyle p^3$, prove that $\displaystyle f(x)$ is irreducible in $\displaystyle K[X]$

=== Supose f(x) is reducible over K ==> K contains a root w (in fact, both) of f(x) (see discussion above), so then F_p(w) <= K. Now use multiplicativity of degrees: [K:F_p] = [K:F_p(w)][F_p(w):F_p]
Note: I think we must add here the assumption that p is an odd prime.

Tonio

This is due in about 12 hours from now, thanks for the help. I'm sorry I left it to the last minute, I tried hard to do it on my own
Even if you only get to this after the deadline I'd still appreciate someone guiding me through this as I want to know this stuff for my midterm next week too
..