Maths Help - Finite Fields

tntcoder · 17 Oct 2009 at 20:13

Hey,

Please can someone help explain the following to me.

Given the field:

It has the following elements in it: 0, 1, 2, x, 2x, x + 1, x + 2, 2x +1 and 2x + 2.

It's also a finite field, because x^2 + 1 is irreducible mod 3.

What I need to understand is how to show that a given polynomial is irreducible in a given field, I don't know how to prove in this case that x^2 + 1 is irreducible.

Also if I was just given a field, how would I find all irreducible polynomials within it?

I'm a tad confused

with my current knowledge I just about know how to work out if something is a field or not, however my polynomial knowledge is pretty limited and I don't really know which techniques i need to be using here.

Thanks for any help.

Gunni Parker · 17 Oct 2009 at 20:14

Are you speaking German???

Manics · 17 Oct 2009 at 20:16

I sometimes wonder if I am of a different race.

regulus · 17 Oct 2009 at 20:17

I can just about add 7 to 12, so I'm afraid I need to pass on this. Good luck though.

Gunni Parker · 17 Oct 2009 at 20:18

Whats a 7??? Good luck dude!

Azagoth · 17 Oct 2009 at 20:18

tntcoder · 17 Oct 2009 at 20:19

Now im really scared

Missing.String · 17 Oct 2009 at 20:31

So what happens if you divide by i then?

frying_pan_cat · 17 Oct 2009 at 20:55

A polynomial is irreducible over a field if it has no roots in that field, so just try putting all of the elements of F_3[X] into your polynomial and show that it's not 0
To find an irreducible polynomial just try one and see if it has any roots. There may be a better way but that's how I'd do it

Hope this helps

DaveF · 17 Oct 2009 at 21:09

frying_pan_cat said:
A polynomial is irreducible over a field if it has no roots in that field,

Not true. If it has no roots in the field then it has no linear factors, but that doesn't rule out it having factors of degree 2 or higher.

e.g. x^4 + 5x^2 + 6 = (x^2+2)(x^2+3) has no real roots but is obviously reducible.

However:

so just try putting all of the elements of F_3[X] into your polynomial and show that it's not 0

is fine here because the poly is quadratic and therefore any factors can only be linear.

To find an irreducible polynomial just try one and see if it has any roots. There may be a better way but that's how I'd do it.

Can't say I see a better plan either, not that I'm great at polys over finite fields.

Note that the OP asked how to find *all* irreducible polys, which I think will be difficult, since I'm pretty sure that if you have a finite field then there are an infinite number of irreducible polynomials.

EffBee · 18 Oct 2009 at 02:00

Reverse the polarity - that usually works on Star Trek.