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Let $R$ be the ring of entire functions $f: \mathbb{C} \rightarrow \mathbb{C}$ that are analytic at every point of $\mathbb{C}$ with respect to point-wise addition and multiplication. Then show that

(A) $R$ is an integral domain

(B) the irreducible elements of $R$ are upto multiplication by units, polynomials of the form $z-\xi$.

(C) $R$ is not a UFD

The first one is fairly easy to do as if $f$ and $g$ are analytic functions such that $fg=0$ on a domain then $f\equiv 0$ or $g \equiv 0$ on some subset of the domain with non-empty interior hence $f\equiv 0$ or $g \equiv 0$ on the domain.

For part $B$ I believe I need a theorem that states in effect that entire functions can be written as product of polynomials $z-\xi$. I need some help with this one and C too. Thanks

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Miz
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  • I believe the Weierstrauss Factorization theorem should give you what you want, with regards to factoring entire functions. – SquirtleSquad Jun 17 '16 at 06:02
  • And for part C, you want to consider the fact that some analytic functions have infinitely man zeroes, so they cannot be written as a finite product. – SquirtleSquad Jun 17 '16 at 06:11
  • Yup thanks. I read the Wikipedia article on Weierstrauss Factorization theorem it helped a lot! – Miz Jun 17 '16 at 06:19

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$f$ is a unit iff it has no zeros. If it has a zero at $z=a$, then $g(z) = f(z)/(z-a)$ has a removable singularity at $a$; after removing that singularity, it is entire. Show that $f$ is irreducible iff $g$ is a unit.

Robert Israel
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