0

Background:

Definition: The function $N:\Bbb{Z}[\sqrt{d}]\to \Bbb{Z}$ given by $$N(s+t\sqrt{d})=(s+t\sqrt{d})(s-t\sqrt{d})=s^2-dt^2$$

is called the norm.

Question:

I would like to know how is the above definition for the norm for quadratic integers interpreted? Is it similar to the Gaussian integers being plotted as vectors on a lattice grid over the complex plane? (visual representation) Also, since norm has to do with distance having to do with vectors, is the idea similar or the same with quadratic integers? As a very naive question, does this notion of the norm for quadratic integers extend to cubic, quartic, etc integer fields? Lastly, are there book references that discuss these type of interpretation at a level that is accessible to someone who have had a course in abstract algebra.

Thank you in advance

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Seth
  • 4,043
  • 1
    In general, if $k\subseteq K$ is a finite dimensional extension of a field, then $K$ is a vector space over $k,$ and the map $p_{\alpha}:K\to K$ for $\alpha\in K$ is defined as $\beta\mapsto \alpha\beta,$ then the norm of $\alpha$ is the determinant of $p_\alpha.$ – Thomas Andrews May 13 '25 at 21:13
  • @ThomasAndrews can you point me to a book reference. If possible, are there any kind of visual representation somewhere. I feel like this definition is screaming for some sort of illustration. – Seth May 13 '25 at 21:18
  • 3
    In your case, there is a standard representation of your ring as matrices: $$a+b\sqrt d\leftrightarrow \begin{pmatrix}a&bd\b&a\end{pmatrix}$$ The norm is the determinant of the matrix. This matrix has two eigenvalues, $a+b\sqrt d$ and $a-b\sqrt d.$ It turns out, these correspond to the two "automorphisms" of $\mathbb Q[\sqrt d],$ and the determinant is the product of those two eigenvalues. – Thomas Andrews May 13 '25 at 21:20
  • 1
    Probably too soon to look into that, because most of the books I know that cover it this way are about field theory, and you are apparently first studying number theory. But just accept it for now as a definition. The Wikipedia page is https://en.wikipedia.org/wiki/Field_norm?wprov=sfti1# – Thomas Andrews May 13 '25 at 21:25
  • There really isn't a good "illustration." Certainly, there is often a geometric meaning to linear algebra, and there, the determinant can represent a measure of the hypervolume, but that seems obscure to me, rather than clarifying. – Thomas Andrews May 13 '25 at 21:27
  • @ThomasAndrews oh this is from linear algebra. Okay okay, that makes sense. But since this can be interpreted using matrices. I am guessing group representations will make an appearance if i pursue this topic far enough? – Seth May 13 '25 at 21:29
  • try this, the last 80 pages are on quadratic forms https://bookstore.ams.org/mbk-105/ – Will Jagy May 13 '25 at 21:29
  • 1
    Well, there is a more general concept of "ring representations," where groups are just a special case. The above correspondence is such a ring representation. Given a group, $G,$ there is a ring $\mathbb C[G],$ and group representations of $G$ correspond to ring representations of the ring. – Thomas Andrews May 13 '25 at 21:33
  • @ThomasAndrews thank you for your answers. For some reasons, I was trying to understand the geometric meaning behind this norm when studying the last bit about quadratic integers in abstract algebra from Dummit and Foote and Hungerford. – Seth May 13 '25 at 21:36
  • @WillJagy thank you for the refference. – Seth May 13 '25 at 21:36
  • 1
    Yeah, when $d=-1,$ the elements of the field are complex numbers, and $N(z)=|z|^2,$ where $|z|$ is the usual complex number absolute value, which makes it feel very geometric. But the geometry is much harder to see in the other cases. The word "norm" is so often used in analysis for a notion of absolute values or distances, but the relationship is not so direct. In particular, algebraic norms can be negative. – Thomas Andrews May 13 '25 at 22:02
  • The norm has a geometric interpretation via the area of a fundamental lattice parallelogram, e.g. see KCd's comment $\ \ $ – Bill Dubuque May 13 '25 at 22:41
  • @BillDubuque just a side question, if i want to count the number of lattice points with a figure in the plane containing either gaussian or quadratic integers, I used tools from geometry of numbers and those trusty estimation tools from asymptotic analysis? – Seth May 14 '25 at 03:25
  • @ThomasAndrews when you say algebraic norm can be negative, are you referring to the notion of valuation in the case of absolute values. I am also wondering what happens for the case of cubic or higher, does the definition change along with how to meaningful interpret the definition geometrically speaking. I know there has been more than ocean filles worth of buckets of books written about the quadratic integers, but what about for the n-th root field of integers?? – Seth May 14 '25 at 03:47
  • Your question about "$n$-th root field of integers" is answered by the very first comment @Thomas posted. – Gerry Myerson May 14 '25 at 04:15
  • 1
    @GerryMyerson thank you for clearing that up. I did not know what Thomas meant by "in gerneral". – Seth May 14 '25 at 06:00

0 Answers0