1
$\begingroup$

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?

  • 0
    I think I got the proof. Thanks a lot.2012-04-04

2 Answers 2

2

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 $U(S)=\mathbb{Q}\setminus\{0\}$.

1

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) $