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cbarker1

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**Problem (c) for Discrete Value Ring for a unit**

I am stuck in the middle of a proof. Here is the background information from Dummit and Foote Abstract Algebra 2nd ed.:

Let $K$ be a field. A discrete valuation on $K$ on a function $\nu$: $K^{\times} \to \Bbb{Z}$ satisfying

- $\nu(a\cdot b)=\nu(a)+\nu(b)$ [i.e. $\nu$ is a homomorphism from the multiplication group of nonzero elements of $K$ to $\Bbb{Z}$]
- $\nu$ is surjective, and
- $\nu(x+y)\ge \min{[\nu(x),\nu(y)]}$, for all $x,y\in K^{\times}$ with $x+y\ne 0$

From part b.) Prove that nonzero element $x\in K$ either $x$ or $x^{-1}$ is in $R$.

Proof: Suppose $x\in K^{\times}$. Assume $x\notin R$. Then $\nu(x)<0$, we have

\begin{align*}

&0=\nu(1)\\

&=\nu(xx^{-1})\\

&=\nu(x)+\nu(x^{-1})

\end{align*}

the last line implies $\nu(x^{-1})>0$. Therefore, $x^{-1} \in R$. QED

Here is the question: Prove that an element $x$ is a unit of $R$ if and only if $\nu(x)=0$.

Proof: Suppose $x$ is a unit of $R$. By part (b), $\nu(x)>0$...

Thanks

CBarker1

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