Write $(x-5, 3)$ as a principle ideal in $\mathbb{Q}[X]$

Asked Nov 25 '22 at 19:44

Active Nov 26 '22 at 06:48

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I'm trying to familiarize myself with ideals and principle ideals but I'm having some problems when it comes to writing principle ideals when they're not explicitly principle. For example I would like to write $(x-5, 3)$ as a principle ideal in $\mathbb{Q}[X]$. I know that it is possible to write $(x-5, 3)=(f)$ for $f\in\mathbb{Q}[X]$ because $\mathbb{Q}[X]$ is a PID, but I can't seem to be able to find such an $f$. I know that $(x-5,3)=\{(x-5)g+3h:g,h\in\mathbb{Q}[X]\}$, but I don't know how this would help.

Any help is greatly appreciated.

asked Nov 25 '22 at 19:44

Num2

2

Hint: this ideal contains an invertible element... – Mark Nov 25 '22 at 19:46
Wouldn't all constant polynomials of $(x-5,3)$ have invertible elements? – Num2 Nov 25 '22 at 20:00
Yes, and when can an ideal contain an invertible element? – Mark Nov 25 '22 at 20:25
More generally $,(a,b) = (b),$ if $,b\mid a,$ (e.g. when $b$ is a unit - which divides everything. This is a special case of ideal mod reduction (when it mods out to $0)\ \ $ – Bill Dubuque Nov 26 '22 at 06:44

Write $(x-5, 3)$ as a principle ideal in $\mathbb{Q}[X]$

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