User:James in dc/sandbox/Divisor theory

In algebraic number theory (and more generally in ring theory), divisor theory describes factorization and divisibility. It extends the concepts of "evenly divides" and "greatest common divisor (gcd)" from the integers (or, nore generally, from any ring where gcd is defined) to the ring of integers in any algebaic extension. This allows a sort of unique factorization: every element of the field is a unique product of prime divisors, but the divisors are not part of the field. In some cases (principal ideal domains) the divisors correspond to field elements: the prime divisors act like prime numbers, there is unique factorization, and the arithmetic is completely analogous to that of $$\mathbb{Q}$$. In the other case, where there are divisors that do not correspond to field elements, it can be proved that factorization is non-unique.

Since they are describing the same phenomena, divisor theory and ideal theory share some notation and terminology.

History
Kummer introduced "ideal complex numbers" in 1846 to describe division in fields without unique factorization. The word "ideal" was in use in e.g., projective geometry, at the time and means that something wasn't "really" present (like the line at infinity) He evidently thought of them as numbers missing from some specific field but being present in a larger field (now known as the Hilbert class field).

Kronecker developed divisor theory in 1882 but only published one paper on it.

Dedekind defined an "ideal" as the set of numbers divisible by a divisor. His approach is now the standard, but some authors prefer divisors Weyl discusses the differences between divisor theory and ideal theory

Notation, conventions
$$\mathbb{Z}\subset \mathbb{Q}$$ are the integers and rational numbers.

$$p, P\in\mathbb{Z}$$ are primes.

If $$r$$ is a ring $$\mathbb{Q}_r$$ is its field of quotients and $$r[x]$$ is the ring of polynomials with coefficients in $$r$$.

Capital Roman letters are fields: $$\mathbb{Q}\subset A,K,L$$

Greek letters are elements of fields: $$\alpha\in L$$

Fraktur letters are divisors: $$\mathfrak{1=a a^{-1}}$$

Integers, units, and divisibility
An element $$\alpha\in A\supset r$$. is algebraic over r if there is a polynomial $$f_{\alpha}(x) \in r[x]$$ such that $$f_{\alpha}(\alpha)=0.$$

If $$f_{\alpha}$$ is monic $$\alpha$$ is an algebraic integer. This is equivalent to saying that some power of $$a$$ is an $$r$$-linear combination of lower powers:
 * $$a^n=r_{n-1}a^{n-1}+r_{n-2}a^{n-2}+\dots+r_{1}a+r_0, \;\;\;r_i\in r.$$

The adjective "algebraic" is often omitted and the numbers in $$\mathbb{Z}$$ referred to as rational integers.

The integers in $$A$$ form an integral domain denoted $$\mathbb{Z}_A,$$ $$[A],$$ or $$\mathfrak{O}_A$$

For $$\alpha,\beta\in A$$ the symbol $$\alpha|\beta$$ is read "$$\alpha$$ (exactly) divides $$\beta$$" and means that $$\frac{\beta}{\alpha}\in\mathfrak{O}_A$$. (Division by 0 is not defined.)

Note that $$1|a$$ is equivalent to $$a\in\mathfrak{O}_A$$ or "$$a$$ is an integer".

A unit $$\eta$$ is any integer that divides 1. Note that $$1|\eta|1$$ is equivalent to "$$\eta$$ is a unit."

Greatest common divisor
For $$\alpha,\beta,\gamma,\delta,\mu \in A$$,

$$\delta$$ is a greatest common divisor (gcd) of $$\alpha$$ and $$\beta$$, written $$\delta=(\alpha,\beta)$$, if


 * 1) $$\delta|\alpha$$ and $$\delta|\beta$$. and


 * 2) $$\gamma|\alpha$$ and $$\gamma|\beta$$ implies $$\gamma|\delta$$.

Similarly, $$\mu$$ is a least common multiple (lcm) of $$\alpha$$ and $$\beta$$, written $$\mu=[\alpha,\beta]$$, if


 * 1) $$\alpha|\mu$$ and $$\beta|\mu$$. and


 * 2) $$\alpha|\gamma$$ and $$\beta|\gamma$$ implies $$\mu|\gamma$$.

$$\alpha$$ and $$\beta$$ are relatively prime (or coprime), written $$\alpha\perp\beta$$, if $$(\alpha,\beta)=1$$.

Natural rings
A ring $$r$$ is natural if


 * 1) it is an integral domain,


 * 2) any two elements have a GCD,


 * 3) given an $$\alpha \in r$$ there is a maximum number of non-unit factors in any factorization $$\alpha =\beta_1\beta_2\dots\beta_n$$, and


 * 4) there is an algorithm for factoring polynomials in $$r[x].$$

A polynomial $$f \in r[x_1, x_2, \dots]$$ is primitive if the GCD of its coefficients is a unit.

The rings $$\mathbb{Z}, \;\mathbb{Q}[x,y,..], \;\mathbb{Z}[x,y,.. ], \;F_q[x,y,..]$$ are natural.

Lemma
In a natural ring if $$c\mid ab$$ but $$c\nmid a$$ and $$c\nmid b$$ then $$c=c_1c_2$$ where neither $$c_1$$ nor $$c_2$$ is a unit.

Proof: Let $$d=(ab,cb).$$ Then $$b|d.$$ Set $$c_1=\frac{d}{b}$$. Then $$c_1|a$$ and $$c_1|c.$$ Let $$c_2=\frac{c}{c_1}$$. Since $$c|ab$$ and $$c|cb,$$ $$c|d.$$ That is $$c_1c_2|bc_1,$$ so $$c_2|b.$$ If $$c_1|1$$ then $$c=c_1c_2|b$$ contrary to assumption. If $$c_2|1$$ then $$c|c_1$$ implying $$c|a$$ contrary to assumption.

Divisors
Throughout this section $$K$$ is an algebraic extension of the natural ring $$r$$.

Divisibility of polynomials and divisors
Let $$f,g\in K[x_1,x_2,\dots]$$ be polynomials over $$K$$ in any number of variables. $$f$$ divides $$g,$$ written $$f|g,$$ if there is a polynomial $$q\in Z_K[x_1,x_2,\dots]$$ such that $$g=fq.$$

The polynomial $$f$$ represents a divisor $$\mathfrak{a},$$ written $$\mathfrak{a}=[f].$$

The divisor represented by $$f$$ divides the polynomial $$g,$$ written $$[f]|g$$ if there is a primitive polynomial $$\pi\in r[x_1,x_2,\dots]$$ such $$f|g\pi$$ i.e. $$fq=g\pi$$ where all the coefficients of $$q$$ are integral over $$r.$$

Let $$ f_1,f_1,g\in K[x_1,x_2,\dots].$$ The divisor represented by $$f_1$$ divides the divisor represented by $$f_2$$, written $$[f_1]|[f_2]$$ if $$f_2|g$$ implies $$f_1|g.$$

$$f_1$$ and $$f_2$$ represent the same divisor, written$$[f1]=[f2],$$ if $$[f_1]|[f_2]$$ and $$[f_2]|[f_1].$$

Lemmas
1) If $$f|g_1$$ and $$f|g_2$$ then $$f|(g_1\pm g_2).$$

2) Let $$g_1, g_2,\dots\in K$$ be the coefficients of $$g\in K[x_1,x_2,\dots].$$ If $$[f]|g_i$$ for every $$i$$ then $$[f]|g.$$

3) If $$[f]|g$$ and $$[g]|h$$ then $$[f]|h.$$

4) $$[f]|g$$ if and only if $$[f]|[g]$$.

5) Let $$g_1, g_2,\dots\in K$$ be the coefficients of $$g\in K[x_1,x_2,\dots].$$ If $$d=(g_1,g_2,\dots)$$ then $$[g]=[d].$$

6) If $$[f_1]|[f_2]$$ then $$[f_1f_3]|[f_2f_3].$$

Divisors
Divisors divide things: algebraic numbers $$\mathfrak{d}|\alpha$$ or other divisors $$\mathfrak{d}|\mathfrak{e}$$. Divisors can be used as moduli $$\alpha\equiv\beta\pmod{\mathfrak{d}}$$ means $$\mathfrak{d}|(\alpha-\beta)$$.

Formally, they are a multiplicative group, traditionally written in fraktur script: $$\mathfrak{1=aa^{-1}}$$.

Divisors are represented by sets of algebraic numbers $$\mathfrak{a}=(\alpha_1,\alpha_2,\dots\alpha_r)$$ is read as "a is the greatest common divisor of the alphas". A singleton $$\mathfrak{r}=(\rho)$$ is read as "r is the "divisor of rho". Zero is ignored $$(\alpha,0,\beta,\dots) = (\alpha,\beta,\dots)$$ and $$(0)$$ is not defined.

Order doesn't matter: $$(\alpha_1,\alpha_2,\dots\alpha_r) =(\alpha_1^',\alpha_2^',\dots\alpha_r^')$$ if $$\alpha_1^',\alpha_2^',\dots\alpha_r^'$$ is a permutation of $$\alpha_1,\alpha_2,\dots\alpha_r$$The product of $$\mathfrak{a}=(\alpha_1,\alpha_2,\dots\alpha_r)$$ and $$\mathfrak{b}=(\beta_1,\beta_2,\dots\beta_s)$$ is $$ \mathfrak{ab}= (\alpha_1\beta_1,\alpha_2\beta_1,\dots\alpha_r\beta_1, \alpha_1\beta_2,\alpha_2\beta_2,\dots\alpha_r\beta_2, \dots\alpha_r\beta_s ) $$

The GCD of $$\mathfrak{a}$$ and $$\mathfrak{b}$$ is given by $$\mathfrak{(a,b)}=(\alpha_1,\alpha_2,\dots\alpha_r, \beta_1,\beta_2,\dots\beta_s)$$

Divisor theory is developed axiomatically.

Two divisors are equal if they divide each other: if $$\mathfrak{a|b }$$ and $$\mathfrak{b|a}$$ then  $$\mathfrak{a=b }$$

The divisor of a number divides the number: $$(\gamma)|\gamma$$

If $$\mathfrak{d}|\gamma$$ and $$\mathfrak{d}|\delta$$ then $$\mathfrak{d}|(\alpha\gamma\pm\beta\delta)$$.

If $$ \alpha=\beta\gamma$$ then $$ (\alpha)=(\beta)(\gamma)$$

If $$\mathfrak{d}|\mathfrak{e}$$ then $$\mathfrak{d}|\mathfrak{ef}$$.

These axioms imply that if $$\alpha$$ is a unit, $$\alpha|1$$, then $$(\alpha)=\mathfrak{1}$$

Let $$A\supset r$$ be an algebraic extension of a natural ring $$r$$. The group of (nonzero fractional) divisors of $$A$$ is an abelian group $$\mathbb{D}_A$$ finitely generated by the set $$\mathbb{P}_A\subset\mathbb{D}_A$$, the primes of $$A$$. For each prime $$p \in \mathbb{Z} $$ there are a finite number (one or more) of primes $$\mathfrak{p}\in\mathbb{P}_A$$. "lying above" $$p$$. The identity of $$\mathbb{D}_A$$ is $$\mathfrak{1}$$.

For any divisor $$\mathfrak{d} \in \mathbb{D}_A$$
 * $$\mathfrak{d} = \mathfrak{p}\mathfrak{q}\dots\mathfrak{s}, \;\;\mathfrak{p},\mathfrak{q},\dots\mathfrak{s}\in\mathbb{P}_A$$

and if $$\mathfrak{d} = \mathfrak{p}\mathfrak{q}\dots\mathfrak{s}=\mathfrak{p'}\mathfrak{q}'\dots\mathfrak{s'}\;\;\;$$ the $$\mathfrak{p'}, \mathfrak{q}'\dots$$, are a permutation of the $$\mathfrak{p}, \mathfrak{q}\dots$$

This can be written as an infinite product
 * $$\mathfrak{d}=

\prod_{ \mathfrak{p}\in \mathbb{P}_A} \mathfrak{p}^{\mathfrak{d}_\mathfrak{p}}. $$

where $$\mathfrak{d}_\mathfrak{p}$$ is an integer. For any $$\mathfrak{d}$$ only a finite number of $$\mathfrak{d}_\mathfrak{p}$$ are nonzero.

The $$\mathfrak{d}_\mathfrak{p}$$ define divisibility for the divisors:



\begin{align}

&\mathfrak{p}=\mathfrak{a}\mathfrak{b}  &&\mathfrak{p}_\mathfrak{p} = \mathfrak{a}_\mathfrak{b} + \mathfrak{b}_\mathfrak{p}\\

&\mathfrak{q}=\mathfrak{a}/\mathfrak{b}  &&\mathfrak{q}_\mathfrak{p} = \mathfrak{a}_\mathfrak{b} - \mathfrak{b}_\mathfrak{p}\\

&\mathfrak{d}=(\mathfrak{a},\mathfrak{b})&&\mathfrak{d}_\mathfrak{p} = \min(\mathfrak{a}_\mathfrak{p},\mathfrak{b}_\mathfrak{p})\\

&\mathfrak{m}=[\mathfrak{a},\mathfrak{b}]&&\mathfrak{m}_\mathfrak{p} = \max(\mathfrak{a}_\mathfrak{p},\mathfrak{b}_\mathfrak{p})\\

&\mathfrak{a}\perp\mathfrak{b}              &&\mathfrak{a}_\mathfrak{p}\mathfrak{b}_\mathfrak{p}=0\\

&\mathfrak{d}=\mathfrak{1}               &&\mathfrak{d}_\mathfrak{p}=0\\

&\mathfrak{a}|\mathfrak{b}               &&\mathfrak{a}_\mathfrak{p}\le\mathfrak{b}_\mathfrak{p}\\

\end{align} $$

Note that these imply that if $$\mathfrak{1|a}$$ then $$\mathfrak{a}_\mathfrak{p} \ge 0$$ (which in some circumstances means that $$\mathfrak{a}$$ is an integer) and that $$\mathfrak{(a,b)[a,b]=ab}$$.

There is a homomorphism $$:A^\times\rightarrow\mathbb{D}_A$$. Its image is the principal subgroup $$\mathbb{H}_A<\mathbb{D}_A$$. The quotient group $$\mathbb{Cl}_A=\mathbb{D}_A/\mathbb{H}_A$$, the class group, is finite.

Most importantly for $$\alpha, \beta\in A^\times$$
 * $$\alpha|\beta$$ if and only if $$(\alpha)|(\beta)$$.

Main theorem
The field of quotients of a natural ring has divisors.

(Kronecker 1882) An algebraic extension of a field with divisors also has divisors. .

Quadratic
$$\mathbb{P}_{\mathbb{Q}}=\{ 2,3,5,7,11,13,17,19\dots\}$$

$$\mathbb{P}_{\mathbb{Q}(i)}=\{(1+i),(3),(2+i),(2-1),(7),(11),(3+2i),(3-2i),(4+1),(4-i),(19)\dots\}$$

$$\mathbb{P}_{\mathbb{Q}(\sqrt{-5})}=\{(2,1+\sqrt{-5}),(3,1+\sqrt{-5}),(3,1-\sqrt{-5}),(\sqrt{-5})),(7s),(11),(13),(17),(19),\dots\}$$

Gauss's lemma
Edwards discusses the history

Gauss
$$f,g\in\mathbb{Q}(x), \;f,g\;$$ monic.

If $$f\not\in\mathbb{Z}(x)$$ or $$g\not\in\mathbb{Z}(x)$$ then $$fg\not\in\mathbb{Z}(x)$$.

Modern (Bourbaki)
$$f,g\in\mathbb{Z}(x)$$. The content of a polynomial $$f,\; \operatorname{ct}(f),\;$$ is the gcd of its coefficients.

$$\operatorname{ct}(fg)=\operatorname{ct}(f)\operatorname{ct}(g)$$

Dedekind
$$f=\sum f_ix^i,\;\;g=\sum g_jx^j\in\mathbb{Q}(x)$$.

If $$fg\in\mathbb{Z}(x)$$ then every product $$f_i g_j\in\mathbb{Z}$$.

Dedekind Prague
$$A$$ is an algebraic number field, $$ \mathbb{Z}_A$$ its ring of integers

$$f=\sum f_ix^i,\;\;g=\sum g_jx^j\in A(x)$$.

If $$fg\in\mathbb{Z}_A(x)$$ then every product $$f_i g_j\in\mathbb{Z}_A$$.

Kronecker
Let $$R=\mathbb{Z}[a_0,a_1,\dots a_m, \;\;b_0,\dots b_n]$$.

Define $$c_0,c_1,\dots c_{m+n}\in R$$ by
 * $$c_i=\sum_{j+k=i} a_jb_k$$

and let
 * $$S=\mathbb{Z}[c_0, c_1,\dots c_{m+n}]\subset R$$.

Each of the $$(m+1)(n+1)$$ products $$( a_jb_k )$$ is integral over $$S$$.


 * $$( a_jb_k )^N=p_0( a_jb_k )^{N-1} +p_1(a_jb_k )^{N-2}+\dots+p_N$$

where $$p_i\in S$$

For example when $$ m=n=1 $$ ($$c_0=a_0b_0, c_1=a_0b_1+a_1b_0,c_2=a_1b_1$$)


 * $$(a_0b_1)^2=c_1(a_0b_1)-c_0c_2$$

Or when $$ m=n=2$$ ($$c_0=a_0b_0, c_1=a_0b_1+a_1b_0,c_2=a_0b_2+a_1b_1+a_2b_0,c3=a_1b_2+a_2b_1,c_4=a_2b_2$$)



\begin{align} (a_0b_1)^6= &(a_0b_1)^5(-3c_1) + \\ &(a_0b_1)^4(-3c_1^2-2c_0c_2)+\\ &(a_0b_1)^3(c_1^3-4c_0c_1c_2) + \\ &(a_0b_1)^2(-c_0^2c_1c_3-2c_0c_1^2c_2-c_0^2c_2^2+4c_0^3c_4)+\\ &(a_0b_1)\;\;(c_0^2c_11^2c_3+c_0^2c_1c_2^3-4c_0^2c_1c_4)+\\ &\;\;\;\;\;\;\;\;\;\;\;\;(c_0^3c_1c_2c_3+c_0^4c_3^2+c_0^3c_1^2c_4) \end{align}$$

Kronecker content theorem
Let $$ g_1, g_2,\dots\in K[x_1,x_2,x_3\dots]$$

The content of a polynomial, $$\mathfrak{J}(f)$$ is the divisor of $$K$$ defined by its coefficients.


 * $$\mathfrak{J} (g_1 g_2,\dots) = \mathfrak{J} (g_1)\mathfrak{J} (g_2)\dots$$