Irreducible Polynomial of Degree 3

[Justabeginner]

Homework Statement

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.

Homework Equations

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.

 

[haruspex]

Is this approach correct?

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

 

[Mark44]

Homework Statement

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.

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.

Homework Equations

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.