An irreducible polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field. In other words, it is a polynomial that cannot be broken down any further.
In abstract algebra, an irreducible polynomial is used to define a cyclic group. Specifically, the roots of an irreducible polynomial over a field form a cyclic group under multiplication. This is known as the Galois group of the polynomial.
No, an irreducible polynomial by definition cannot be factored into lower degree polynomials. This means that it cannot have multiple roots, as that would require it to be able to be broken down into lower degree polynomials.
To determine if a polynomial is irreducible, you can use the Eisenstein's criterion, which states that if a polynomial has a prime number that divides all but the leading coefficient and the constant term, and the prime number squared does not divide the constant term, then the polynomial is irreducible.
Irreducible polynomials and cyclic groups have numerous applications in mathematics and computer science. For example, they are used in coding theory to construct error-correcting codes, in cryptography to generate random numbers, and in number theory to study prime numbers. They are also important in algebraic geometry and algebraic topology.