How can I prove that there is an infinite amount of irreducible polynomials $\pmod 2$?
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How do you prove that there is an infinite amount of primes? – player3236 Sep 13 '20 at 13:45
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https://math.stackexchange.com/questions/1222025/there-are-infinitely-many-irreducible-polynomials-in-bbbf-px – Representation Sep 13 '20 at 13:47
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Here, Number of monic irreducible polynomials of prime degree p over finite fields, is discussed Gauss's formula for counting the irreducible polynomials of a general degree $n\gt 0$ over a finite field. So since there are irreducible polynomials of every degree, there are infinitely many of them. – hardmath Sep 13 '20 at 13:56
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Hint: $\mathbb{F}_p[x]$ is UFD. Do you know the Euclid’s proof of infinitude of primes?
Seewoo Lee
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If there are fintely many, multiply them together and add 1. This polynomial is not divisible by any of the finite set. So it should have a new irreducible factor.
In fact there is an irreducible polynomial of any degree $>0$.
markvs
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