# Problem (c) for Discrete Value Ring

• MHB
• cbarker1
Therefore $x$ is a unit.In summary, the conversation discusses the concept of discrete valuation on a field and its properties, as well as a proof regarding the element $x$ being a unit of $R$ if and only if $\nu(x)=0$. The proof involves utilizing the properties of $\nu$ and deducing the condition for $x$ to be a unit.
cbarker1
Gold Member
MHB
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
1. $\nu(a\cdot b)=\nu(a)+\nu(b)$ [i.e. $\nu$ is a homomorphism from the multiplication group of nonzero elements of $K$ to $\Bbb{Z}$]
2. $\nu$ is surjective, and
3. $\nu(x+y)\ge \min{[\nu(x),\nu(y)]}$, for all $x,y\in K^{\times}$ with $x+y\ne 0$
The set $R=\left\{x\in K^{\times}| \nu(x)\ge 0\right\} \cup \left\{0\right\}$.
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

Last edited:
If $x$ is a unit of $R$, then $\nu(x),\, \nu(x^{-1}) \ge 0$; since $\nu(x) + \nu(x^{-1}) = 0$ one deduces $\nu(x) = 0$. Conversely, if $\nu(x) = 0$, then by the equation $\nu(x) + \nu(x^{-1}) = 0$ it follows that $\nu(x^{-1}) = 0$. Thus $x^{-1}\in R$.

## 1. What is a Discrete Value Ring?

A Discrete Value Ring is a mathematical structure that consists of a set of discrete values and two operations, addition and multiplication, that follow certain rules. It is similar to a ring in abstract algebra, but the values are discrete rather than continuous.

## 2. What is "Problem (c) for Discrete Value Ring"?

"Problem (c) for Discrete Value Ring" is a specific problem or question related to the properties and operations of a Discrete Value Ring. It could refer to a specific theorem or concept that needs to be proven or understood.

## 3. How is a Discrete Value Ring different from a traditional ring?

A Discrete Value Ring differs from a traditional ring in that it has a finite number of discrete values, whereas a traditional ring can have an infinite number of values. Additionally, the operations of addition and multiplication in a Discrete Value Ring may follow different rules than in a traditional ring.

## 4. What are some real-world applications of Discrete Value Rings?

Discrete Value Rings have applications in various fields such as computer science, cryptography, and coding theory. They can be used to model discrete systems and are also important in the study of finite fields.

## 5. How can one solve "Problem (c) for Discrete Value Ring"?

The solution to "Problem (c) for Discrete Value Ring" will depend on the specific problem and the properties of the Discrete Value Ring. One approach could be to use the axioms and properties of a Discrete Value Ring to prove the desired result or to find a counterexample that disproves the statement.

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