In an article on 'Comparing the close cousins $\mathbb{Z}$ and $\mathbb{F}_q[x]$', I've found the following

The fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials in $\mathbb{C}[x]$. Since every polynomial of degree $n$ over $\mathbb{C}$ has $n$ roots in $\mathbb{C}$, we see that every polynomial over $\mathbb{C}$ factors completely into linear polynomials, that is, the only irreducibles in $\mathbb{C}$ are the linear polynomials. By using a similar reasoning, over $\mathbb{R}$, $f(x)$ factors into a product of linear and/or quadratic polynomials; that is, the irreducibles over $\mathbb{R}$ are either linear or quadratic. Thus we see that, in some sense, irreducibles in $\mathbb{C}[x]$ and $\mathbb{R}[x]$ are relatively scarce, and thus it is quite difficult to solve an analogue to the Goldbach Conjecture.

My Question

I don't quite get the last statement. Why can we say that the irreducibles are relatively scarce? Can't there still be almost infinitely many linear and/or quadratic irreducible polynomials? Why can the article state that 'thus it is quite difficult to solve an analogue to the Goldbach Conjecture'?

  • 2
    $\begingroup$ We'd have to figure out what the analogue is first. It evidently can't be "every polynomial is the sum of two irreducible polynomials". $\endgroup$ – Dylan May 15 '18 at 8:48
  • 3
    $\begingroup$ Maybe the author meant "it is quite difficult to pose a plausible analogue to the Goldbach Conjecture." $\endgroup$ – lulu May 15 '18 at 9:09

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