Find all units of $S$, where $S$ is the set of polynomials in $\mathbb{Q}[x]$ whose coefficient of $x$ is $0$. I think the units are $\mathbb{Q} \setminus \{0\}$. Is that correct?
Polynomials (abstract algebra)
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abstract-algebra
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0I think I got the proof. Thanks a lot. – 2012-04-04
2 Answers
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We have $R=\mathbb{Q}[x]$ and $S=\mathbb{Q}[x^2,x^3]$. Since $U(R)=\mathbb{Q}\setminus\{0\}\subseteq S$, it follows that $U(S)=\mathbb{Q}\setminus\{0\}$.
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Hint $\ $ Let $\rm\:T = \mathbb Q[x].\:$ By $\rm\:U(T) = U(\mathbb Q)\:$ and unit inheritance \rm\:R\subset R'\:\Rightarrow\:U(R)\subset U(R')\: follows
$\rm\: \mathbb Q\subset S\subset T\ \ \Rightarrow\ \ U(\mathbb Q) \subset U(S) \subset U(T)\subset U(\mathbb Q)\:\ \Rightarrow\ \ U(\mathbb Q) = U(S) = U(T) $