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 $P(0)=-1$?

  • $\begingroup$ I would assume you could look at polynomials of the form $x^i(x-1)^j(x+1)^k -1$ for different $i,j,k$. $\endgroup$ – Arthur Dec 4 '16 at 18:02
  • $\begingroup$ Hint: How would you prove there are infinitely many prime natural numbers $p$ such that $p\equiv 2\pmod {3}$? $\endgroup$ – Thomas Andrews Dec 4 '16 at 18:05

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

  • $\begingroup$ Hey, you already wrote this...I shall delete my answer now. +1 $\endgroup$ – DonAntonio Dec 4 '16 at 18:10
  • $\begingroup$ And then continue similar to Euclid's proof? $\endgroup$ – ChikChak Dec 4 '16 at 18:12
  • $\begingroup$ @Thomas Andrews $\endgroup$ – ChikChak Dec 4 '16 at 20:08

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