1. ## Irreducibility

I have an exam tomorrow, and I'm a little confused on the idea of irreducibility of polynomials. Now, a root is something where if you plug it in you get 0 right? I also have the definition of irreducible but I still don't get it. So if someone could give me an example or something to show me what it means that would be great.

2. First, a complaint about terminology. Many people write just as you do- that a "root" of a polynomial makes that polynomial 0. Strictly speaking a "zero" of a polynomial is a value of the variable that makes the polynomial equal to 0. A "root" of an equation is a value of the variable that satisfies the equation. But, so many people confuse the two, I'm probably fighting a losing battle!

A Polynomial is "irreducible" over over a field (typically the rational numbers or some extension of the rational numbers) (of course, it must have coefficients in that field) if it cannot be factored into factors with coefficients in the field. Since if a is a zero of a polynomial, then the polynomial has x- a as a factor, that is exactly the same as saying that a polynomial is irreducible if it has no zeros in the field.

For example $\displaystyle x^2- 2$ is irreducible over the rational numbers because it cannot be factored as (x-a)(x-b) for any rational a and b.
It is, however, reducible over $\displaystyle Q(\sqrt{2})$, the smallest field containing all rational numbers and $\displaystyle \sqrt{2}$ because that field contains $\displaystyle \sqrt{2}$ which is a zero. And that means it can be factored as $\displaystyle (x-\sqrt{2})(x+\sqrt{2})$.

3. Originally Posted by Janu42
I have an exam tomorrow, and I'm a little confused on the idea of irreducibility of polynomials. Now, a root is something where if you plug it in you get 0 right? I also have the definition of irreducible but I still don't get it. So if someone could give me an example or something to show me what it means that would be great.
Let R be an integral domain and cosider the Ring of polynomials

$\displaystyle R[x]$

Case I: If R is the the rational numbers

$\displaystyle x^2-2$ is irreducable in $\displaystyle \mathbb{Q}[x]$

This can be checked by rational roots theorem (the only choices are $\displaystyle \pm 1, \pm 2$) or note that it is eisenstien with p=2

Case II: if R is the real numbers then the polynomial is reduceable because

$\displaystyle \sqrt{2}$ is a roots and $\displaystyle \sqrt{2} \in \mathbb{R}$

Note that the rational roots theorem only works for degree < 3 polynomials.

for instance

in $\displaystyle \mathbb{Q}[x]$

$\displaystyle x^4-4x^2+4$ have has no rational roots but it is reduceable because if can be factored into two elements, with each factor not being a unit.

$\displaystyle (x^2-2)(x^2-2)$.

Irreducability depends of your field of coeffeints.