Pinter's textbook "A book of abstract algebra" asks to prove the following:     There are infinitely many irreducible polynomials in $\mathbb{Z}_{5} [x]$.     Here's my attempt to prove it any field $\mathbb{F}$:  Let $F$ be a field. Suppose that there are finitely many irreducible polynomials in $F[x]$, say, $r_1 , r_2 , \ldots , r_n $. Then $r_1 r_2 \ldots r_n +1 \in F[x]$. Hence, by the factorisation into irreducible polynomial theorem tells us that  $$r_1 r_2 \ldots r_n +1 = k( r_1 r_2 \ldots r_n)$$  for some $k \in \mathbb{F}$. But then $r_1 r_2 \ldots r_n$ would be an associate of $1$. Thus each of $r_i$ must be a constant polynomial which contradicts the definition of a irreducible polynomial.    But I suppose this is not what the author wanted. I was wondering if there's any way to generate such irreducible polynomials over $\mathbb{Z}_{5}$. It looks to me as if $x^n+1$ where $n$ is even is a irreducible polynomial but I couldn't prove it. Any hints?
I can intuitively understand that there are infinitely many irreducible polynomials in $\Bbb{F}_p[X]$, where $p$ is a prime number, but I'm having a hard time actually proving it. What proof strategy should I take for this?
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Here's the loop I'm using:  &lt;?php $paged = get_query_var('paged') ? get_query_var('paged') : 1; $args  = array(     'posts_per_page' =&gt; 4,     'orderby'        =&gt; 'modified',     'post_type'      =&gt; "projekte",     'paged'          =&gt; $paged ); $custom_query = new WP_Query($args); $c = 0; while($custom_query-&gt;have_posts()) :     $custom_query-&gt;the_post();     ?&gt;     &lt;article&gt;...&lt;/article&gt;     &lt;?php endwhile; wp_reset_postdata(); // reset the query ?&gt;   I've gone through the functions.php a million times looking for a some limiting get_pre_posts filter but there's nothing I can see.  I also got an error log setup and nothing is returned there, I've tested it that it works.  Any suggestions? Design problem - Implement an approval process on case, which sends email to approver showing some additional info, which should not be visible to case owner/agent (who clicked on "Submit for approval" button).  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