Help finding units in a ring

In summary, units in a ring are elements that have a multiplicative inverse and play a crucial role in the structure and properties of the ring. They can be determined by checking for a multiplicative inverse or if they are relatively prime to the ring's order. A ring can have multiple units, and they are closely related to the concept of a ring homomorphism, particularly unit-preserving homomorphisms.
  • #1
Jupiter
46
0
How can I show that if
[tex]\frac{a}{a^2-2b^2},\frac{b}{a^2-2b^2}\in \mathbb{Z}[/tex]

then [tex] a^2-2b^2=\pm 1[/tex]?
If you care to see the whole problem, you can find it here:
http://www.math.rochester.edu/courses/236H/home/hw12.pdf [Broken]
It's #4 part c.

BTW, why is the significance of this "norm map"? I tried to google it for fun, but couldn't find much.
 
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  • #2
I got it! I needed part b. I was going about the problem the wrong way. Thanks anyway!
 
  • #3


To show that a^2-2b^2=\pm 1, we can use the fact that the elements a and b are units in the ring \mathbb{Z}, which means they have multiplicative inverses. This means that there exist elements c and d such that ac=ca=1 and bd=db=1. We can then rewrite the given expressions as:

\frac{a}{a^2-2b^2} = \frac{1}{a-2b^2a} = c
\frac{b}{a^2-2b^2} = \frac{1}{a-2b^2a} = d

Multiplying these expressions together, we get:

\frac{ab}{(a^2-2b^2)^2} = cd

Since cd=1, we can rewrite this as:

\frac{ab}{(a^2-2b^2)^2} = 1

This can be rearranged to give us:

a^2-2b^2 = \pm 1

This shows that if the given expressions are elements of \mathbb{Z}, then a^2-2b^2 must equal \pm 1.

Now, for the significance of the "norm map", it is a function that maps elements of a ring to the set of integers. In this case, the norm map is defined as:

N(a+bi) = a^2-2b^2

This map helps us understand the structure of the ring \mathbb{Z} (the Gaussian integers, which are numbers of the form a+bi, where a and b are integers). By using the norm map, we can show that the elements a and b are units in the ring if and only if N(a+bi)=\pm 1. This is exactly what we showed above, which is why the norm map is significant in this problem.
 

1. What are units in a ring?

Units in a ring refer to elements that have a multiplicative inverse, meaning they can be multiplied by another element in the ring to give the identity element. In other words, they have a reciprocal that when multiplied together, results in 1.

2. Why is it important to find units in a ring?

Finding units in a ring is important because they play a crucial role in the structure and properties of the ring. They help determine the divisibility and invertibility of elements, and are essential in solving equations and proving theorems in ring theory.

3. How do you determine if an element is a unit in a ring?

To determine if an element is a unit in a ring, you can check if it has a multiplicative inverse. This can be done by finding the greatest common divisor (GCD) of the element and the ring's identity element. If the GCD is 1, then the element is a unit. Another method is to check if the element is relatively prime to the ring's order.

4. Can a ring have multiple units?

Yes, a ring can have multiple units. In fact, every ring has at least two units - the identity element and its additive inverse. Some rings, such as fields, have all elements except 0 as units. Other rings, such as the integers, only have ±1 as units.

5. How are units related to the concept of a ring homomorphism?

Units are closely related to ring homomorphisms. A ring homomorphism is a function that preserves the ring structure, meaning it maps the ring's operations of addition and multiplication. If a ring homomorphism maps a unit to a unit, then it is called a unit-preserving homomorphism. This property is important in studying the behavior of units in different rings.

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