monic irreducible polynomials

Question

I need help with :

Let $\Bbb F_3=\Bbb Z_3$ be the field with 3 elements.Show that there are infinitely many monic irreducible polynomials in $\Bbb F_3[x]$ such that $P(0)=-1$.

Now,I saw this proof Ring of polynomials over a field has infinitely many primes but I am not sure,is there any different for this field and with the condition that $P(0)=-1$?

I would assume you could look at polynomials of the form $x^i(x-1)^j(x+1)^k -1$ for different $i,j,k$. — Arthur, Dec 04 '16 at 18:02
Hint: How would you prove there are infinitely many prime natural numbers $p$ such that $p\equiv 2\pmod {3}$? — Thomas Andrews, Dec 04 '16 at 18:05

score 2 · Accepted Answer · answered Dec 04 '16 at 18:07

2

Given finitely many prime polynomials $p_1,p_2,\dots,p_n\in\mathbb F_3[x]$, consider that $xp_1(x)p_2(x)\cdots p_n(x)-1$.

answered Dec 04 '16 at 18:07

Thomas Andrews

177,126

Hey, you already wrote this...I shall delete my answer now. +1 – DonAntonio Dec 04 '16 at 18:10
And then continue similar to Euclid's proof? – ChikChak Dec 04 '16 at 18:12
@Thomas Andrews – ChikChak Dec 04 '16 at 20:08

monic irreducible polynomials

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