puzzled about a statement
 

Hi, I'm reading something about Quadratic Fields from Hardy's "An introduction to the theory of numbers".

I've come to an impasse. There's a sentence which I've outlined in black which I don't understand. It says "is plainly equivalent to" but I simply can't how the statement about the norm is equivalent to the version of the division algorithm (with the inequality of Norms) above. I'd really appreciate some help making sense of it if possible.



[URL]https://www.mersenneforum.org/attachment.php?attachmentid=19195&stc=1&d=1541016049[/URL]

[QUOTE=wildrabbitt;499190]
I've come to an impasse. There's a sentence which I've outlined in black which I don't understand. It says "is plainly equivalent to" but I simply can't how the statement about the norm is equivalent to the version of the division algorithm (with the inequality of Norms) above. I'd really appreciate some help making sense of it if possible.
[/QUOTE]

Hint: [TEX]\delta=\gamma/\gamma_1[/TEX].

Also remember that N(αβ)=N(α)N(β).

Thanks to both of you.

I'm nearly ready to move on after a whole day going crazy.

[QUOTE=wildrabbitt;499286]I'm nearly ready to move on after a whole day going crazy.[/QUOTE]
Mathematics research often feels like that.
Your tenacity is a good thing! :smile:

[attach]19212[/attach]

next thing
 

Hi again,

I've been reading on a bit in the book and there's something else I don't get.
If I could have another hint or a clear explanation I'd be really grateful.


[URL]https://www.mersenneforum.org/attachment.php?attachmentid=19223&stc=1&d=1541344310[/URL]

I've outlined something with a black line. The (14.7.2) refers to the inequality with this number on the page from my first post.

Firstly, I don't see why [TEX]r=\frac{1}{2}[/TEX] and [TEX]s=\frac{1}{2}[/TEX] have been chosen as special because I think all possible numbers from the quadratic field (ie any r and s) aught to be under consideration.

Secondly, I don't see why chosing these values for r and s leads to the simple

inequality [TEX]\frac{1}{2} + \frac{1}{2}\mu < 1[/TEX]. What happens to [TEX]x^2, -x, y^2 and -y[/TEX]?


It'd be nice if someone could point the way for me.

BTW, I don't know how to embed an image into a post. If someone could explain it'd make things better.

Sorry I've made a mistake. I was looking at 14.7.1 rather than 14.7.2 .Should be okay now.

Actually, I didn't make a mistake. My mistake was thinking I'd made one.

[QUOTE=wildrabbitt;499548]
Firstly, I don't see why [TEX]r=\frac{1}{2}[/TEX] and [TEX]s=\frac{1}{2}[/TEX] have been chosen as special because I think all possible numbers from the quadratic field (ie any r and s) aught to be under consideration.

Secondly, I don't see why chosing these values for r and s leads to the simple

inequality [TEX]\frac{1}{2} + \frac{1}{2}\mu < 1[/TEX]. What happens to [TEX]x^2, -x, y^2 and -y[/TEX]?


It'd be nice if someone could point the way for me.
.[/QUOTE]
We are assuming that the quadratic field is Euclidean so, indeed, for any rational r and s there must exist integers x and y such that
\[ |(r-x)^2-m(s-y)^2|<1.\]
In particular, this must hold if we take \(r=s=\frac{1}{2}\).
Putting \(\mu=-m>0\), there must exist integers x and y satisfying
\[ \left(\frac{1}{2}-x\right)^2+\mu\left(\frac{1}{2}-y\right)^2<1.\]
But for [B]all[/B] integers x we have \((\frac{1}{2}-x)^2\geq\frac{1}{4}\) and similarly for y ...

[QUOTE=wildrabbitt;499548]
BTW, I don't know how to embed an image into a post. If someone could explain it'd make things better.[/QUOTE]
Once you have uploaded an image, you can insert it in your text by clicking the arrow next to the paper clip and choosing it from the drop-down menu.
But don't put too many full pages of Hardy & Wright on this public forum or we'll get intro copyright trouble! :smile:

Thanks Nick.

I see now. That's very helpful.

Is it possible for the pages to be deleted. I don't mind.

[QUOTE=Nick;499560]
But don't put too many full pages of Hardy & Wright on this public forum or we'll get intro copyright trouble! :smile:[/QUOTE]

or Just write in TeX .