How should I prove this question?

Prove that any two integers that are associates have the same norm.

Printable View

- Oct 25th 2006, 10:51 AMbeta12a prove question
How should I prove this question?

Prove that any two integers that are associates have the same norm. - Oct 25th 2006, 12:27 PMtopsquark
- Oct 25th 2006, 12:29 PMThePerfectHacker
What does it mean associates?

According to PlanetMath and Wikipedia associates are the units of the ring with unity. But what ring are you in? Surly (Captain**Blank**) word not in $\displaystyle \mathbb{Z}$ where the associates are -1 and 1. Are you in the ring of quadradic integers? - Oct 25th 2006, 12:38 PMbeta12
Thanks topsquark.

Hi perfecthacker,

Yes, in quadratic integers.

Here is the definition of associate :

Two quadratic intergers alpha and beta are associates provided that the quotient alpha/beta is a unit.

That mean N(alpha/beta) = 1 or -1 - Oct 25th 2006, 12:54 PMtopsquark
Interesting. My book does talk about associates being defined on an integer ring (which is why I was thinking there might be a typo in your question...that you weren't talking about integers, but two elements of an integer ring), but my book doesn't define them as being units.

The reason I didn't mention my thoughts about the "typo" is that two associates (as defined in my book) don't necessarily have the same norm.

-Dan - Oct 25th 2006, 01:12 PMbeta12
The reason I didn't mention my thoughts about the "typo" is that two associates (as defined in my book) don't necessarily have the same norm.

-Dan[/QUOTE]

How does your book define associates?

That is interesting issue.

The conclusion which I gave are according to my book and lectures. I am sure no " typo " in it.

===============

Also, perfecthacker,

I am so sorry for the double post. I did really forget. - Oct 25th 2006, 05:18 PMtopsquark
The book is "Algebra" by Thomas Hungerford, Chapter 3, section 3, pg. 135.

"A nonzero element a of a commutative ring R is said to divide an element $\displaystyle b \in R$ (notation: a|b) if there exists $\displaystyle x \in R$ such that ax = b. Elements a, b of R are said to be associates if a|b and b|a."

In an earlier section (Chapter 3, section 1, pg. 116) it defines a unit*:

"An element a in a ring R with identity is said to be left invertible if there exists $\displaystyle c \in R$ such that $\displaystyle ca = 1_R$. The element c is called a left inverse of a. An element $\displaystyle a \in R$ that is both left and right invertible is said to be invertible or to be a unit."

* I edited out the right invertible definition to make this easier to read. You can supply that extension easily enough.

Clearly my book is making a distinction between the definitions of unit and associate. And I note that in $\displaystyle \mathbb{Z}_6$ that 2 and 4 are associates, but clearly not units. (According to the above definitions.) However these are general definitions and not specific to quadratic rings. (To which I've forgotten the definition of, and my book doesn't mention.) Perhaps that makes the difference.

-Dan - Oct 26th 2006, 05:18 AMbeta12
I see. ;)