# Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

# Splitting of Primes in Extensions

In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.

Let $\mathfrak{p}$ be a prime ideal of the ring of integers $\mathcal{O}_{K}$ of an algebraic number field $K$ (we will sometimes also refer to $\mathfrak{p}$ as a prime ideal of $K$ – this is common practice and hopefully will not cause any confusion). In an algebraic number field $L$ which contains $K$ (we also say that $L$ is an extension of $K$, and write $L|K$), this prime ideal $\mathfrak{p}$ decomposes into a product of prime ideals $\mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r}$ in $\mathcal{O}_L$, with respective exponents $e_{1},e_{2}...e_{r}$, i.e.

$\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}$.

The exponents $e_{1},e_{2}...e_{r}$ are called the ramification indices of the prime ideals $\mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}$. If $e_{i}=1$, and the residue field extension $\kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p})$ (see below) is separable, we say that $\mathfrak{P}_{i}$ is unramified over $K$. If $e_{1}=e_{2}=...e_{r}=1$, we say that the prime $\mathfrak{p}$ is unramified. If all primes of $K$ are unramified in $L$, we say that the extension $L|K$ is unramified.

In the rest of this post we will continue to assume the factorization of $\mathfrak{p}$ as shown above. The residue fields $\kappa(\mathfrak{P}_{i})$ and $\kappa(\mathfrak{p})$ of $\mathcal{O}_{L}$ and $\mathcal{O}_{K}$ at the primes $\mathfrak{P}_{i}$ and $\mathfrak{p}$ are defined as the quotients $\mathcal{O}_{L}/\mathfrak{P}_{i}$ and $\mathcal{O}_{K}/\mathfrak{p}$, and the inertia degrees $f_{i}$ are defined as the degrees of the fields $\kappa(\mathfrak{P}_{i})$ with respect to the field $\kappa(\mathfrak{p})$ (i.e. the dimensions of the vector spaces $\kappa(\mathfrak{P}_{i})$ over the field of scalars $\kappa(\mathfrak{p})$), i.e.

$\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})]$.

The ramification indices $e_{i}$, the inertia degrees $f_{i}$, and the degree $n=[L:K]$ of the field extension $L$ with respect to $K$ are related by the following “fundamental identity“:

$\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n$

In order to understand these concepts better, we can look at the following “extreme” cases:

If $e_{i}=1$ and $f_{i}=1$ for all $i$, then $r=n$, and we say that the prime $\mathfrak{p}$ splits completely in $L$.

If $r=1$ and $f_{1}=1$, then $e_{1}=n$, and we say that the prime $\mathfrak{p}$ ramifies completely in $L$.

If $r=1$ and $e_{1}=1$, then $f_{1}=n$, and we say that the prime $\mathfrak{p}$  is inert in $L$.

Consider for example, the field $\mathbb{Q}(i)$ as a field extension of the field $\mathbb{Q}$. The ring of integers of $\mathbb{Q}(i)$ is the ring of Gaussian integers $\mathbb{Z}[i]$ (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of $\mathbb{Q}$ is the ring of ordinary integers $\mathbb{Z}$. The degree $[\mathbb{Q}(i):\mathbb{Q}]$ is equal to $2$. The prime ideal $(5)$ of $\mathbb{Z}$ splits completely as the product $(2+i)(2-i)$ in $\mathbb{Z}[i]$, the prime ideal $(2)$ of $\mathbb{Q}$ ramifies completely as $(1+i)^{2}$ in $\mathbb{Z}[i]$, while the prime ideal $(3)$ of $\mathbb{Z}$ is inert in $\mathbb{Z}[i]$.

We now bring in Galois groups. We assume that $L$ is a Galois extension of $K$. This means that the order of $G(L|K)$, the Galois group of $L$ over $K$, is equal to the degree of $L$ over $K$. In this case, it turns out that we will have

$\displaystyle e_{1}=e_{2}=...=e_{r}$

and

$\displaystyle f_{1}=f_{2}=...=f_{r}$.

The fundamental identity then becomes

$efr=n$.

This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal $\mathfrak{P}$ of $\mathcal{O}_{K}$, we define the decomposition group $G_{\mathfrak{P}}$ as the subgroup of the Galois group $G$ that fixes $\mathfrak{P}$, i.e.

$\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}$.

The elements of $L$ that are fixed by the decomposition group $G_{\mathfrak{P}}$ form what is called the decomposition field of $K$ over $\mathfrak{P}$, denoted $Z_{\mathfrak{P}}$:

$\displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}$

Every element $\sigma$ of $G_{\mathfrak{P}}$ automorphism $\bar{\sigma}$ of $\kappa(\mathfrak{P})$ sending the element given by $a\text{ mod }\mathfrak{P}$ to the element given by $\sigma a\text{ mod }\mathfrak{P}$. The residue field of the decomposition field $Z_{\mathfrak{P}}$ with respect to $\mathfrak{p}$ is the same as the residue field of the field $K$ with respect to $\mathfrak{p}$, which is $\kappa(\mathfrak{p})$. Therefore we have a surjective homomorphism

$\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$

which sends the element $\sigma$ of $G_{\mathfrak{P}}$ to the element $\bar{\sigma}$ of $G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$. The kernel of this homorphism is called the inertia group of $\mathfrak{P}$ over $K$. Once again, the elements of $L$ fixed by the inertia group $I_{\mathfrak{P}}$ form what we call the inertia field of $K$ over $\mathfrak{P}$, denoted $T_{\mathfrak{P}}$:

$\displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}$

The groups $G_{\mathfrak{P}}$, $I_{\mathfrak{P}}$, $G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$ are related by the following exact sequence:

$\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0$

Meanwhile, the relationship between the fields $K$, $Z_{\mathfrak{P}}$, $T_{\mathfrak{P}}$, and $L$ can be summarized as follows:

$\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L$

The ramification index, inertia degree, and the number of primes in $K$ into which a prime $\mathfrak{p}$ in $L$ splits are given in terms of the degrees of the aforementioned fields as follows:

$\displaystyle e=[L:T_{\mathfrak{P}}]$

$\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]$

$\displaystyle r=[Z_{\mathfrak{P}}:K]$

Let $\mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}$, and $\mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}$. We also refer to $\mathfrak{P}_{Z}$ (resp. $\mathfrak{P}_{T}$) as the prime ideal of $Z_{\mathfrak{P}}$ (resp. $T_{\mathfrak{P}}$) below $\mathfrak{P}$.

The ramification index of $\mathfrak{P}$ over $\mathfrak{P}_{T}$ is equal to $e$, and its inertia degree is equal to $1$. Meanwhile, the ramification index of $\mathfrak{P}_{T}$ over $\mathfrak{P}_{Z}$ is equal to $1$, and its inertia degree is equal to $e$. Finally, the ramification index and inertia degree of $\mathfrak{P}_{Z}$ over $\mathfrak{p}$ are both equal to $1$.

We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let $\mathfrak{P}$ be a prime that is unramified over $K$. Then $G_{\mathfrak{P}}$ is isomorphic to $G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$, it is cyclic, and it is generated by the unique automorphism

$\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}$    for all    $\displaystyle a\in \mathcal{O}_{K}$

where $q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]$. The automorphism $\varphi_{\mathfrak{P}}$ is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.

References:

Splitting of Prime Ideals in Galois Extensions on Wikipedia

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Number Fields by Daniel Marcus

Algebraic Theory of Numbers by Pierre Samuel

Algebraic Number Theory by Jurgen Neukirch