Abstract-Irreducible plynomials over Q

[WannaBe22]

Homework Statement

Prove that the next polynomials are irreducible over Q:
B. 8x^3 -6x -1

Homework Equations

The Attempt at a Solution

I'll be glad to receive some assistance

Thanks a lot!

 

[VeeEight]

For B, does the polynomial have roots in Q? Use the fact that the polynomial is of degree 3 (if a polynomial of degree 2 or 3 has no roots in F then...)

 

[WannaBe22]

How can I show the specific plynomial has no roots in Q?
And I can't understand how to continue the theorem you gave...
About the other parts-as you can see, I've managed to solve them...
I only need help in B...

Hope you'll be able to continue helping me...

Thanks a lot

 

[JSuarez]

The solution can be simple, if you covered Gauss' lemma (which states that if the polynomial is irreducible over the integers, then it's also irreducible over the rationals), or quite messy, if you have to prove directly.

 

[WannaBe22]

I've learned thic specific lemma...I know that if a polynomial is irreducible over Z then it's also irreducible over Q...But how can I prove it's irreducible over Z?

 

[VeeEight]

And I can't understand how to continue the theorem you gave...

See: 4.2.7. Proposition in http://www.math.niu.edu/~beachy/aaol/polynomials.html

(the proof is pretty simple and involves simply factoring a polynomial of degree 2 or 3 and finding a root in F)

 

[WannaBe22]

Ok...But how can I prove that this polynomia has no root over Q?
Is there no simple way involving Eisenstein's criterion or something?

TNX

 

[Dick]

Ok...But how can I prove that this polynomia has no root over Q?
Is there no simple way involving Eisenstein's criterion or something?

TNX

Use the Rational Roots Theorem.

 

[WannaBe22]

Hmmmm So if k/l is a root of this polynomial then k|(-1) and l | 8
The possibilities are: k=1,-1 , l=+-1,+-2,+-4,+-8

Should I check for each one of the 16 possibilities that it isn't a root?

Thanks

 

[Dick]

Well, I count 8 possibilities, but sure. It doesn't take long to run through them.

 

[WannaBe22]

Yep...8 possibilities-sry...
Well, let's see:
p(1)=1, p(-1)=-3, p(0.5)=-3, p(-0.5)=1, p(0.25)=-19/8, p(-0.25)=3/8,p(1/8)=-111/64
p(-1/8)=-17/64 ... So we have no rational roots to this polynomial and then it's irreducible :)
About the propisition: 4.2.7. Proposition. A polynomial of degree 2 or 3 is irreducible over the field F if and only if it has no roots in F.
If we take a polynomial of degree 2-it must be decomposite into two factors of degree 1 and each and every one of them is a root...When we take a polynomial of degree 3, it can only be reduced into 2-1 factors or 1-1-1 and we must have one factor that defines a root...

I'll be glad to receive some verification!

Thanks a lot to all of you!

 

[Dick]

That's it. It isn't until you get to degree 4 where a polynomial can have two quadratic factors, neither of which have any roots over Q.

 

[WannaBe22]

Thanks a lot! You've all been very helpful!