What are the irreducible elements of the ring of entire functions ? I cannot find these. please anybody help.
Finding irreducible elements of the ring of entire functions
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abstract-algebra
complex-analysis
ring-theory
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0Well, entire functions with exactly one zero of order one are irreducible: if $f$ is such a function and $f(z) = g(z)h(z)$ is a factorization into entire functions, then exactly one of $g(z)$ and $h(z)$ has a zero where $f$ does, and the other one has no zeros and is therefore a unit. Are these all of them? – 2017-01-19
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0They are exactly those of the form $f(z)(z-a)$ where $a \in \Bbb{C}$ and $f(z)$ is any entire nonvanishing function. – 2017-01-19
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1Please write the complete proof .thanks. – 2017-01-20
1 Answers
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An irreducible element $f$ must not be invertible, so it must have a zero.
Let $a$ be a zero of $f$; then $f(z)=(z-a)g(z)$, with $g$ entire. What should $g$ be if $f$ is irreducible?
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1g must be non vaniahing over C.Is it? – 2017-01-21
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0@mathislove It must be invertible, yes. – 2017-01-21