Let $k$ be a field and let $R$ be the subring of $k[x]$ consisting of all polynomials having no linear term, that is, $f (x) \in R$ if and only if $f (x) = s_0 +s_2x^2 +s_3x^3 +\ldots + s_nx^n$.
Show that $x^5$ and $x^6$ have no gcd in $R$.
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Please see https://math.meta.stackexchange.com/questions/5020/ – Angina Seng Aug 28 '19 at 04:58
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Hint only as this sounds like a homework question.
- Suppose there is a gcd $g(x)=g_0+g_2x^2+g_3x^3+\cdots+g_nx^n$.
- Explain why $x^2\mid g(x)$ and $x^3\mid g(x)$.
- Find out what you can about the coefficients $g_0,\ldots\,$.
- Obtain a contradiction.
David
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