# Homework Help: Irreducible Polynomial of Degree 3

1. Aug 20, 2014

### Justabeginner

1. The problem statement, all variables and given/known data
If p(x) ∈F[x] is of degree 3, and p(x)=a0+a1∗x+a2∗x2+a3∗x3, show that p(x) is irreducible over F if there is no element r∈F such that a0+a1∗r+a2∗r2+a3∗r3 =0.

2. Relevant equations

3. The attempt at a solution
Is this approach correct?
If p(x) is reducible, then there exists ax + b such that a, b ε F and a≠0. And p(x) = (ax + b)(cx^2 + dx + e). Then an r exists such that p(r) = 0.

Thank you.

2. Aug 22, 2014

### haruspex

Yes, though you could go into a bit more of an explanation as to why there would have to be a first degree factor.

3. Aug 22, 2014

### Staff: Mentor

With over 300 posts in this forum, you should have learned enough of the ropes here to write exponents clearly.

At the very least, use ^ to indicate exponents, as you do below. Even nicer would be to use the exponent button from the advanced menu - click Go Advanced to show this menu, and then click the X2 to make exponents.

I think what you meant above was p(x) = a0 + a1x + a2x2 + a3x3 = 0, and similarly for your other equation.