Explaining what it means for a polynomial to be irreducible over F Let F be a field and let F[x] be the ring of polynomials in one indeterminate
x with coefficients in F. Explain what it means for a polynomial g(x) in F[x] to be irreducible over
F.
I am thinking like this (very important to get it right):
It means that the polynomial of degree n>=1 with coefficients in a Field F is said to be irreducible over F if it cannot be written as a product of two non-constant polynomials over F of degree less than n.
Enough to get it correct? Or maybe add a line about them all being primes?
 A: You don't need to state that it has a factorization into primes with at least two factors, but it's equivalent for polynomial rings.
In a general ring, an element $r$ is irreducible if, whenever it is written as $r=ab$, exactly one of $a$ and $b$ is invertible. Writing $r\mid a$ if there exists $b$ such that $r=ab$, another way to say this is that $r$ is irreducible if, whenver $a\mid r$, $r\mid a$.
This is different to prime, which is: $r$ is prime if $r$ is not zero of invertible and, whenever $r\mid ab$, $r\mid a$ or $r\mid b$.
In an integral domain (with unity) $R$, every prime is irreducible, but not every irreducible need be prime. To see that every prime is irreducible, suppose that $r$ is prime and $r=ab$. Since $r=ab$, certainly $r\mid ab$. Thus $r\mid a$ or $r\mid b$, without loss of generality $r\mid a$. Thus $a=rs$ for some $s$, and thus
$$ r=ab=rsb.$$
Since $R$ is an integral domain and $r(1-sb)=0$, we see that $sb=1$. In particular, $b$ is invertible, so that $r$ is irreducible.
Rings where every irreducible is prime are unique factorization domains, or UFDs.  This is equivalent to the statement that every element $r\in R$ has a factorization into irreducibles, and, the irreducibles that appear in this factorization are unique up to reordering and multiplying by invertible elements.
Polynomial rings (in potentially many variables) over fields are examples of UFDs, so for polynomial rings there are multiple potential definitions of irreducible.
