Irreducible polynomials over a field F play a similar in a c

Irreducible polynomials over a field F play a similar, in a certain sense, role in F[x] to the role the prime numbers play for Z. Using your favorite proof of the existence of an infinite number of primes as inspiration, please prove that there are infinitely many irreducible polynomials over a field F.

Solution

if there were finite such polynomials.... p1.p2.p3.........pn

then

p1 * p2 * p3 * ......pn +1 (multiplicative unity) is a polynomial

which is either irreducible or can be written as a product of irreducible polynomials.

It cannot be irreducible since it is different from the set of irreducibles.

Also it cannot be divisible by any irreducible . Say it does pi divides this new polynomial fr some i , then

pi divides p1*p2*......pn + 1

and also pi is one of p1,p2....pn so divides p1*p2*....pn

hence it divides 1

contradiction