Why should I care about proving a polynomial to be irreducible?

Question

Are there any number theoretic/combinatorial/other applications of proving that a polynomial is irreducible over integers/any other field ?(But integers are preferred)

Answer 1

A basic use of this concept: a polynomial $p(x)$ is irreducible over a field $F$ if and only if $F[x]/p(x)$ is a field.

More generally, suppose you want to understand the isomorphism class of the quotient $F[x]/p(x)$ for an arbitrary polynomial $p(x)$. This boils down to understanding the prime factorization of $p(x)$; say it is

$$p(x) = \prod_i p_i(x)^{m_i}$$

where the $p_i$ are irreducible. Then

$$F[x]/p(x) \cong \prod_i F[x]/p_i(x)^{m_i}$$

by the Chinese remainder theorem, where $F[x]/p_i(x)^{m_i}$ is an artinian local ring with residue field $F[x]/p_i(x)$. Geometrically we are trying to understand the "scheme of zeroes" $\text{Spec } \mathbb{F}[x]/p(x)$ of $p(x)$, and the theorem above tells us that this scheme consists topologically of a finite set of ("fat") points, one for each irreducible factor of $p(x)$.

Irreducibility is a very fundamental concept, and if you keep studying abstract algebra and/or number theory you'll keep running into it sooner or later. There's no need to be in any particular hurry, though; either you'll run into it or you won't.