Ring Theory: Proving $\mathbb{Z} [ \sqrt{2} ]$ has Infinitely Many Units

In summary, we are discussing the set of elements \mathbb{Z} [ \sqrt{2} ] and how to find an inverse for each element in order to show that there are infinitely many units. We initially thought that the only units were 1 and -1, but it turns out there are more and we are trying to find a way to generate more units using a simple method.
  • #1
QuantumJG
32
0
Show [tex] \mathbb{Z} [ \sqrt{2} ] [/tex] = [tex] \{ a + b \sqrt{2} | a,b \in \mathbb{Z} \} [/tex] has infinitely many units.

I started by taking an element:

[tex] a + b \sqrt{2} \in \mathbb{Z} [ \sqrt{2} ] [/tex]

and finding an inverse

[tex] \left( a + b \sqrt{2} \right) ^{-1} [/tex]

such that the product gives zero and tried to show any element works. But I'm not sure about doing this.
 
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  • #2
The product should give 1.
 
  • #3
And I found the inverse and I didn't see an infinite number of units. Z is the integers, right? What did you get for the inverse?
 
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  • #4
Dick said:
And I found the inverse and I didn't see an infinite number of units. Z is the integers, right? What did you get for the inverse?

Ooops. My mistake. There are more units than just 1 and -1. Can you find some? Once you've found one that isn't 1 or -1, can you think of a simple way to use it to generate more?
 
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FAQ: Ring Theory: Proving $\mathbb{Z} [ \sqrt{2} ]$ has Infinitely Many Units

What is Ring Theory?

Ring Theory is a branch of abstract algebra that studies algebraic structures called rings. A ring is a set with two operations, addition and multiplication, that satisfy certain properties.

What is the significance of proving that $\mathbb{Z} [ \sqrt{2} ]$ has Infinitely Many Units?

Proving that $\mathbb{Z} [ \sqrt{2} ]$ has Infinitely Many Units is significant because it confirms that this particular ring has an infinite number of elements that have multiplicative inverses. This means that there are infinitely many ways to combine these elements to obtain the identity element, which is essential in many mathematical applications.

How is the proof for $\mathbb{Z} [ \sqrt{2} ]$ different from other proofs of infinite units in different rings?

The proof for $\mathbb{Z} [ \sqrt{2} ]$ is different from other proofs of infinite units in other rings because it utilizes the unique properties of this specific ring. In particular, it uses the fact that $\sqrt{2}$ is an irrational number and that the ring only contains elements of the form $a + b\sqrt{2}$, where $a$ and $b$ are integers. These specific properties allow for a simpler and more direct proof.

Can this proof be applied to other rings?

Yes, this proof can be applied to other rings with similar properties. For example, it can be used to prove that rings of the form $\mathbb{Z} [ \sqrt{n} ]$, where $n$ is a positive integer that is not a perfect square, also have infinitely many units. However, it may not be applicable to other rings with different properties.

Are there any practical applications for this proof?

Yes, there are practical applications for this proof in various fields of mathematics, including number theory and algebraic geometry. It also has applications in cryptography, as it helps to understand and analyze certain encryption algorithms. Additionally, the concept of units in rings is fundamental in many areas of abstract algebra and has implications in other areas of mathematics and science.

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