Unique Factorization

Question

I need help with the following:

(i) Show that in R{x}, no polynomial of odd degree >1 is irreducible.

The question is a bit confusing because I interpret it as Show that in R{x}, all polynomials of odd degree >1 are reducible.

For example:

$$x^5 + x^4 + 1$$ should be reducible but I'm not sure if it is, how do I know if it is reducible or not?

(ii) Show that if f(x) in R{x} has a multiple factor, then its derivative f'(x) is not relatively prime to f(x).

It's reducible over $\mathbb R$, it's not reducible over $\mathbb Z$. Ask yourself why $x^3-2$ is reducible over $\mathbb R$, first. Then $x^5-2$, etc. — Thomas Andrews, Dec 05 '13 at 02:20
I see, the section of our book did not mention the IVT so I assumed we couldn't use it. — Benji_Bombadill, Dec 05 '13 at 02:34
are you familiar with something like every complex polynomial has a complex root... I doubt you are not :O — , Dec 05 '13 at 02:40

score 2 · Answer 1 · answered Dec 05 '13 at 02:22

2

Hint: Any odd degree polynomial has a root, which implicitly gives a factorization over $\mathbb R$.

answered Dec 05 '13 at 02:22

Thomas Andrews

score 1 · Accepted Answer · answered Dec 05 '13 at 02:25

1

For (ii), write $f=g^2h$ with $g$ not constant. Then $f'= 2gg'h+g^2h'$. Hence $g$ divides both $f$ and $f'$.

answered Dec 05 '13 at 02:25

lhf

2 Answers2