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April 3rd, 2018, 08:41 AM   #21
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Quote:
Originally Posted by zylo View Post
a+b$\displaystyle \sqrt{d}\equiv$ (a,b)
What exactly does this mean? The left hand side of this is an element of $\mathbb{Z}[\sqrt{d}]$ but the right hand side is not.

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Originally Posted by zylo View Post
Why do you suddenly bring up the reals? I have said again and again fields are ruled out of the discussion. It makes ab=1 a totally different situation.
I'm only bringing the reals up to explain why $\mathbb{Z}[\sqrt{d}]$ is an ordered set. $\mathbb{R}$ is an ordered set, so any subset of $\mathbb{R}$ is also an ordered set.

Since $d$ is positive, $\mathbb{Z}[\sqrt{d}]$ is a subset of $\mathbb{R}$, so it is an ordered set.
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April 4th, 2018, 08:48 AM   #22
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What does $\displaystyle \equiv$ mean? If you don't like this notation, forget it. Irrelevant.

What does $\displaystyle a+b\sqrt{d} > e+ f\sqrt{d}$ mean?
What does $\displaystyle a+b\sqrt{d}$ is positive mean?


Quote:
Originally Posted by zylo View Post
Found an answer in BM.

ab=1 -> |ab|= |a||b|=1
|a|$\displaystyle \geq$1 and |b|$\displaystyle \geq1\rightarrow $|a|=|b|=1

But there is a catch: OP properties don't let you prove there is no element between 0 and 1.
You either fudge it, "obviously there is no integer between 0 or 1," or you have to invoke the Well Ordering Principle for Integers: every subset of positive integers contains a least member, to show it.
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Originally Posted by zylo View Post
To show there is no integer between 0 and 1, assume there is. Then by well ordering the set of integers less than 1 has a least member $\displaystyle 0<m<1$. Then $\displaystyle 0<m^{2}<m$ is a contradiction.
EDIT
Or are you saying the d in $\displaystyle Z(\sqrt{d})$ are ordered? So what?, any subset of R is ordered.

Last edited by zylo; April 4th, 2018 at 09:01 AM.
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April 4th, 2018, 10:09 AM   #23
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Quote:
Originally Posted by zylo View Post
What does $\displaystyle a+b\sqrt{d} > e+ f\sqrt{d}$ mean?
What does $\displaystyle a+b\sqrt{d}$ is positive mean?
Is this a serious question? $a + b\sqrt{d}$ is just some real number - we know what it means to say that it's positive or that it's bigger than another real number.

Here's a trick: if I give you any positive integer $d$ and any integers $a$ and $b$, you can check if $a + b \sqrt{d}$ is positive by typing it into a calculator. For example, $1 + 2 \sqrt{3} \approx 4.46$ is positive, while $4 - 3\sqrt{5} \approx - 2.71$ is negative.

$a + b \sqrt{d} > e + f \sqrt{d}$ if and only if $(a - e) + (b - f) \sqrt{d}$ is positive.

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Originally Posted by zylo View Post
EDIT
Or are you saying the d in $\displaystyle Z(\sqrt{d})$ are ordered?
I don't even understand what this means, so probably not!

Quote:
Originally Posted by zylo View Post
So what?, any subset of R is ordered.
$\mathbb{Z}[\sqrt{d}]$ is a subset of R since all its elements are real numbers. Therefore it is ordered according to what you've just said (which is also what I've been saying the past god knows how many posts lol).

Last edited by cjem; April 4th, 2018 at 10:39 AM.
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April 4th, 2018, 01:21 PM   #24
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To be specific:
$\displaystyle Z[\sqrt{2}]= a+b\sqrt{2}$ defines an ordered pair (a,b) just as a+ib defines an ordered pair (a,b).

(a,b)+(c,d) = (a+b,c+d), etc
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April 4th, 2018, 02:20 PM   #25
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Quote:
Originally Posted by zylo View Post
To be specific:
$\displaystyle Z[\sqrt{2}]= a+b\sqrt{2}$ defines an ordered pair (a,b) just as a+ib defines an ordered pair (a,b).

(a,b)+(c,d) = (a+b,c+d), etc
If you want to treat elements of $\mathbb{Z}[\sqrt{d}]$ as ordered pairs in this way, multiplication would be $(a,b) . (a', b') = (aa' + dbb', ab' + a'b)$, and the ordering would be $(a,b) < (a',b')$ if and only if (the real number) $a + b \sqrt{d}$ is less than (the real number) $a' + b' \sqrt{d}$.
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April 5th, 2018, 06:56 AM   #26
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The literal evaluation of a+b$\displaystyle \sqrt{d}$ does not give a member of Z[$\displaystyle \sqrt{d}$] so does not qualify for ordering, which is the same reason that |z|, z complex, doesn't. Please, just look it up.

Apparently you are unaware that an answer to OP has long since been found:

Quote:
Originally Posted by zylo View Post
Found an answer in BM. (Theorem 5)
ab=1 -> |ab|= |a||b|=1
|a|$\displaystyle \geq$1 and |b|$\displaystyle \geq1\rightarrow $|a|=|b|=1
Also in BM
To show there is no integer between 0 and 1, assume there is. Then by well ordering principleof positive integers the set of integers less than 1 has a least member $\displaystyle 0<m<1$. Then $\displaystyle 0<m^{2}<m$ is a contradiction. (Theorem 3)

Z[$\displaystyle \sqrt{2}$] is given as an example of an integral domain, NOT ORDERED, which has non-unit divisors of 1. Standard example by the way. pg 2

BM Birkhoff MacLane, A Survey of Modern Algebra 3rd Ed
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April 5th, 2018, 09:44 AM   #27
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Quote:
Originally Posted by zylo View Post
Z[$\displaystyle \sqrt{2}$] is given as an example of an integral domain, NOT ORDERED, which has non-unit divisors of 1. Standard example by the way. pg 2

BM Birkhoff MacLane, A Survey of Modern Algebra 3rd Ed
Look at exercise 7 of section 1.3 in BM (page 11 in my copy). It asks you to show that $\mathbb{Z}[\sqrt{2}]$ satisfies the definition of an ordered domain.

Now, unless you think this exercise is a trick question, perhaps it's time to rethink your position.
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April 9th, 2018, 09:13 AM   #28
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Quote:
Originally Posted by cjem View Post
Look at exercise 7 of section 1.3 in BM (page 11 in my copy). It asks you to show that $\mathbb{Z}[\sqrt{2}]$ satisfies the definition of an ordered domain.

Now, unless you think this exercise is a trick question, perhaps it's time to rethink your position.
In my copy it's exercise 8 of section 1.3:

"Define "positive" element in the domain $\displaystyle Z[\sqrt{2}]$, and show that the addition, multiplication and trichotomy laws hold."


Also from same reference:
"An integral domain D is said to be ordered if there are certain elements of D, called the positive elements, which satisfy the addition, multiplication and trichotomy laws for integers." my underlining.

"For a given integer a, one and only one of the following alternatives holds: either a is positive, or a=0, or -a is positive."

Fair enough. Frankly I had given up on the thread. Glad I came back. That's an excellent (educational) question and I would like to know the answer. I note positive is in quotes.
When I see an answer I will rethink my postion.

Off hand, the only thing that occurs to me is a+b$\displaystyle \sqrt{2}$ is positive iff a is positive. But that defines a PROPERTY of the element, P(z)=a, rather than the element itself, as positive, which is not strictly the definition of positive, it is "positive," so it is a trick question (which may have been the intent) and my position is unchanged. I note that if this were true, the same would hold for complex numbers, which do not have positive members.

But I welcome another answer.
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April 9th, 2018, 11:11 AM   #29
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Quote:
Originally Posted by zylo View Post
In my copy it's exercise 8 of section 1.3:

"Define "positive" element in the domain $\displaystyle Z[\sqrt{2}]$, and show that the addition, multiplication and trichotomy laws hold."


Also from same reference:
"An integral domain D is said to be ordered if there are certain elements of D, called the positive elements, which satisfy the addition, multiplication and trichotomy laws for integers." my underlining.

"For a given integer a, one and only one of the following alternatives holds: either a is positive, or a=0, or -a is positive."

Fair enough. Frankly I had given up on the thread. Glad I came back. That's an excellent (educational) question and I would like to know the answer. I note positive is in quotes.
When I see an answer I will rethink my postion.

Off hand, the only thing that occurs to me is a+b$\displaystyle \sqrt{2}$ is positive iff a is positive. But that defines a PROPERTY of the element, P(z)=a, rather than the element itself, as positive, which is not strictly the definition of positive, it is "positive," so it is a trick question (which may have been the intent) and my position is unchanged. I note that if this were true, the same would hold for complex numbers, which do not have positive members.

But I welcome another answer.
Note that your idea (to say $a + b \sqrt{d}$ is "positive" if $a$ is positive) won't work: for example, both $1 - 10 \sqrt{d}$ and $1 + 10 \sqrt{d}$ would be "positive", but their product is $(1 - 100d) + 0 \sqrt{d}$, which is not "positive" (as $d \geq 1$ means $1 - 100d$ is not positive). This means products of "positive" elements needn't be "positive", even though they must be in an ordered domain.

Recall: $d$ is positive and squarefree (i.e. the only square number dividing $d$ is $1$).

Now, the easy and enlightening answer is: say $a + b \sqrt{d} \in \mathbb{Z}[\sqrt{d}]$ is "positive" if $a + b \sqrt{d}$ is a positive real number. Since $\mathbb{Z}[\sqrt{d}]$ is a subdomain of the ordered domain of real numbers, this makes everything work (cf exercise 9, section 1.3).

But if you don't like that for some reason, we can do things directly and without referencing $\mathbb{R}$ (but it will take a lot more work). I'll start things off for you.

____________________________________

Say $a + b \sqrt{d} \in \mathbb{Z}[\sqrt{d}]$ is "positive" if either:
(i) $a > 0$ and $b > 0$
(ii) $a > 0$, $b \leq 0$ and $a^2 > b^2 d$
(iii) $a \leq 0, b > 0$ and $a^2 < b^2 d$

(Aside: these are precisely the conditions for $a + b\sqrt{d}$ to be positive as a real number - I didn't just pluck them out of thin air!)

Let's check that this definition makes $\mathbb{Z}[\sqrt{d}]$ into an ordered domain.

1) Check that the sum of two "positive" numbers is "positive".

Let $a + b \sqrt{d}$, $a' + b' \sqrt{d}$ be "positive". If both of them satisfy (i) above, then so does their sum, so it is positive.

Suppose $a + b \sqrt{d}$ satisfies (i) (so $a, b > 0$) and $a' + b' \sqrt{d}$ satisfies (ii) (so $a' > 0, b' \leq 0$ and $(a')^2 > (b')^2 d$). Then their sum is $(a + a') + (b + b') \sqrt{d}$. Since $a > 0$ and $a' > 0$, certainly $a + a' > 0$.

From here there are two possibilities: either $|b| > |b'|$ or $|b| \leq |b'|$. In the first case, $b + b' > 0$ and the sum $(a + a') + (b + b') \sqrt{d}$ satisfies (i), so is positive. In the second case, notice that

$\begin{align*}
(a+a')^2 - (b+b')^2 d &= a^2 + 2aa' + (a')^2 - b^2 d - 2bb'd - (b')^2 d \\
&> a^2 + 2aa' - b^2 d - 2bb'd \qquad \qquad \qquad \qquad &\text{(as $\: (a')^2 > (b')^2 d \:$ by assumption)} \\
&= |a|^2 + 2 |a| |a'| + 2 |b| |b'| d - |b|^2 d &\text{(as $ \:a, a', b > 0$, $b' < 0$)} \\
&> 2|b|d (|b'| - |b|) &\text{(as $\: |a|^2 + 2 |a| |a'| > 0$)} \\
&\geq 0 &\text{(as we're in the case $ \: |b| \leq |b'|)$}
\end{align*}$

This shows $(a + a')^2 > (b+b')^2$, so the sum $(a + a') + (b + b') \sqrt{d}$ satisfies (ii) and so is positive.

Continue this way: for every other combination of (i), (ii), (iii) $a + b\sqrt{d}$ and $a' + b'\sqrt{d}$ might satisfy, check that their sum also satisfies one of them. I won't bother as it is quite time consuming, but I encourage you to have a go!

2) Check that the product of two "positive" numbers is positive.

Same deal here: check that no matter what combination of (i), (ii), (iii) $a + b\sqrt{d}$ and $a' + b'\sqrt{d}$ could satisfy, their product satisfies one of them.

3) Check that every element $a + b\sqrt{d}$ of $\mathbb{Z}[\sqrt{d}]$ satisfies precisely one of the following: $a + b \sqrt{d} = 0$ (i.e. $a = b = 0$), $a + b \sqrt{d}$ is "positive" or $-(a + b \sqrt{d}) = -a - b \sqrt{d}$ is "positive".

As $0 = 0 + 0 \sqrt{d}$ does not satisfy the definition of "positive", it suffices to show the following two statements:
(a) if $a + b \sqrt{d}$ is non-zero and not positive, then $-a -b\sqrt{d}$ is positive
(b) if $a + b \sqrt{d}$ is positive, then $-a -b\sqrt{d}$ is not positive

Let's start with (a):
Suppose $a + b\sqrt{d} \neq 0$ satisfies none of (i), (ii), (iii). i.e. the three following statements hold:

(i)' $a \leq 0$ or $b \leq 0$
(ii)' $a \leq 0$, $b > 0$ or $a^2 \leq b^2 d$
(iii)' $a > 0$, $b \leq 0$ or $a^2 \geq b^2 d$

But this means the following hold
(i)'' $-a \geq 0$ or $-b \geq 0$
(ii)'' $-a \geq 0$, $-b < 0$ or $a^2 \leq b^2 d$
(iii)'' $-a < 0$, $-b \geq 0$ or $a^2 \geq b^2 d$

Now $a + b\sqrt{d} \neq 0$ combined with (i)'' implies $-a > 0$ or $-b > 0$.

If $-a > 0$ and $-b > 0$, then $-a - b\sqrt{d}$ satisfies (i) and we're done.
Else $-a \leq 0$ or $-b \leq 0$.

If $-a \leq 0$, then $-b > 0$ (see the paragraph above starting with "Now"), and now by (ii)'', $a^2 \leq b^2 d$. But this inequality must be strict as $d$ is assumed squarefree. So $-a - b\sqrt{d}$ satisfies (ii) in this case.

The last possibility is that $-b \leq 0$. In this case, we must have $-a > 0$, and then (iii)''' implies $a^2 \geq b^2 d$. Again, $d$ squarefree means this inequality is strict, so $-a - b\sqrt{d}$ satisfies (iii).

So in every case, $-a -b \sqrt{d}$ satisifes one of (i), (ii), (iii), so is positive. This shows (a).

Now for (b):
A similar kind of argument works. Feel free to do this if you're still not convinced.

____________________________________
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April 10th, 2018, 06:37 AM   #30
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cjem points out that a+b$\displaystyle \sqrt{d}$ $\displaystyle \epsilon$ Z$\displaystyle [\sqrt{d}]$ can be ordered by calling those members positive that evaluate to a real positive number
For example, 5-3$\displaystyle \sqrt{2}$ "=," ie, evaluates to, 1.414, which is positive.
Addition, multiplication, and trichotomy follow trivially.

cjem also noted Z$\displaystyle [\sqrt{d}]$ was ordered because it was a subset of the real numbers. I never felt the breeze.

My apologies, and thanks to cjem for his persistence.

The next question, well ordering, is relevant to OP. If 1 is an integer, the OP has been answered utilizing the well-ordering property of the integers.

But what if 1$\displaystyle \equiv$(1,0) is instead the unit of Z$\displaystyle [\sqrt{d}]$?
Are the only units of Z$\displaystyle [\sqrt{d}]$ (divisors of 1) $\displaystyle \pm$1? No, as cjem showed.

Last edited by zylo; April 10th, 2018 at 06:54 AM.
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