Dependencies

Let $K,F$ be number fields with $F\subseteq K$. Let $A$ be an additive subgroup of $K$. Let

and

The relative different ${\mathfrak{d}}_{K/F}$ of $K/F$ is then defined as $(({\mathcal{O}}_K{)}^{\ast }{)}^{-1}$ which one checks is an integral ideal in ${\mathcal{O}}_K$. This is also the annihilator of ${\mathrm{\Omega }}_{{\mathcal{O}}_K/{\mathcal{O}}_F}^1$ if we want to be fancy.

A prime number $p$ is called regular if it does not divide the class number of $\mathbb{Q}({\zeta }_p)$.

Let $A,K$ be commutative rings with $K$ and $A$-algebra. let $B=\{b_1,\dots ,b_n\}$ be a set of elements in $K$. The discriminant of $B$ is defined as

Let $p$ be an odd regular prime. Then

has no solutions with $x,y,z\in \mathbb{Z}$ and $xyz\ne 0$.

Let $K/F$ be a Galois extension of number fields whose Galois group $\mathrm{Gal}(K/F)$ is cyclic with generator $\sigma$. If $\alpha \in K$ is such that $N_{K/F}(\alpha )=1$, then

for some $\beta \in {\mathcal{O}}_K$.

Let $u\in F$ with $F=\mathbb{Q}({\zeta }_p)$ be a unit such that

for some $a\in {\mathcal{O}}_F$. The either $u={ϵ}^p$ for some $ϵ\in {\mathcal{O}}_F^{\times }$ or $p$ divides the class number of $F$.

Let $K/F$ be an extension of number fields and let $S$ denote the set of $\alpha \in {\mathcal{O}}_K$ be such that $K=F(\alpha )$. Then

where $m_{\alpha }$ is denotes the minimal polynomial of $\alpha$.

Let $K/F$ be an extension of number fields with ${\mathfrak{p}}_F,{\mathfrak{p}}_K$ prime ideas in $K,F$ respectively, with ${\mathfrak{p}}_K^e\parallel {\mathfrak{p}}_F$ for $e>0$. Then ${\mathfrak{p}}_K^{e-1}\mid {\mathfrak{d}}_{K/F}$ and ${\mathfrak{p}}_K^e\mid {\mathfrak{d}}_{K/F}$ iff ${\mathfrak{p}}_F\mid e$.

Let $K$ be a number field and $B,B^{\prime }$ bases for $K/\mathbb{Q}$. If $P$ denotes the change of basis matrix, then

Let ${\zeta }_p$ be a $p$-th root of unity for $p$ an odd prime, let ${\lambda }_p=1-{\zeta }_p$ and $K=\mathbb{Q}({\zeta }_p)$. Then

Let $p$ be an prime number, ${\zeta }_p$ a $p$-th root of unity and $\alpha \in \mathbb{Z}[{\zeta }_p]$ with $\alpha =\sum _i{a}_i{\zeta }_p^i$. Let us suppose that there is $i$ such that $a_i=0$. If $n$ is an integer that divides $\alpha$ in $\mathbb{Z}[{\zeta }_p]$, then $n$ divides each $a_i$.

Let $K$ be a number fields and $\alpha \in K$. Then there exists a $n\in \mathbb{Z}\mathrm{\setminus }\{0\}$ such that $n\alpha$ is an algebraic integer.

Let $p$ be an prime number, ${\zeta }_p$ a $p$-th root of unity and $\alpha \in \mathbb{Z}[{\zeta }_p]$. Then ${\alpha }^p$ is congruent to an integer modulo $p$.

Let $p\ge 3$ be an prime number, ${\zeta }_p$ a $p$-th root of unity and $\alpha \in \mathbb{Z}[{\zeta }_p]$. Let $x$ and $y$ be integers such that $x+y{\zeta }_p^i=u{\alpha }^p$ with $u\in \mathbb{Z}[{\zeta }_p{]}^{\times }$ and $\alpha \in \mathbb{Z}[{\zeta }_p]$. Then there is an integer $k$ such that

Let $p\ge 5$ be an prime number, ${\zeta }_p$ a $p$-th root of unity and $x,y\in \mathbb{Z}$ coprime.

For $i\ne j$ with $i,j\in 0,\dots ,p-1$ we can write

with $u$ a unit in $\mathbb{Z}[{\zeta }_p]$. From this it follows that the ideals

are pairwise coprime.

Let $p$ be a regular odd prime, $x,y,z,ϵ\in \mathbb{Z}[{\zeta }_p]$, $ϵ$ a unit, and $n\in {\mathbb{Z}}_{\ge 1}$. Assume $x,y,z$ are coprime to $(1-{\zeta }_p)$ and that $x^p+y^p+ϵ(1-{\zeta }_p{)}^{pn}{z}^p=0$. Then each of $(x+{\zeta }_p^{i}y)$ is divisible by $(1-{\zeta }_p)$ and there is a unique $i_0$ such that $(x+{\zeta }_p^{i_0})$ is divisible by $(1-{\zeta }_p{)}^2$.

Let $p$ be a regular odd prime, $x,y,z,ϵ\in \mathbb{Z}[{\zeta }_p]$, $ϵ$ a unit, and $n\in {\mathbb{Z}}_{\ge 1}$. Assume $x,y,z$ are coprime to $(1-{\zeta }_p)$, $x^p+y^p+ϵ(1-{\zeta }_p{)}^{pn}{z}^p=0$ and $x+y$ is divisible by ${\mathfrak{p}}^2$ and $x+{\zeta }_p^{k}y$ is only divisible by $\mathfrak{p}=(1-{\zeta }_p)$ (for $0<k\le p-1$). Let $\mathfrak{m}=\gcd ((x),(y))$. Then:

1. We can write

   $$(x+y)={\mathfrak{p}}^{p(m-1)+1}\mathfrak{m}{\mathfrak{c}}_0$$

   and

   $$(x+{\zeta }_p^{k}y)=\mathfrak{p}\mathfrak{m}{\mathfrak{c}}_k$$

   (for $0<k\le p-1$) where $m=n(p-1)$ and with ${\mathfrak{c}}_i$ pairwise coprime.

2. Each ${\mathfrak{c}}_k={\mathfrak{a}}_k^p$ and ${\mathfrak{a}}_k{\mathfrak{a}}_0^{-1}$ is principal (as a fractional ideal).

Let $K/F$ be a Galois extension of $F=\mathbb{Q}({\zeta }_p)$ with Galois group $\mathrm{Gal}(K/F)$ cyclic with generator $\sigma$. Then there exists a unit $\eta \in {\mathcal{O}}_K$ such that $N_{K/F}(\eta )=1$ but does not have the form $ϵ/\sigma (ϵ)$ for any unit $ϵ\in {\mathcal{O}}_K$.

Let $L/K$ be an extension of fields and let $B=\{b_1,\dots ,b_n\}$ be a $K$-basis of $L$. Then $\mathrm{\Delta }(B)\ne 0$.

Let $K$ be a number field, $p$ be a rational prime below $\mathfrak{p}$ and let ${\alpha }_0,\xi$ be in ${\mathcal{O}}_{\mathfrak{p}}^{\times }$ (the units of the ring of integers of the completion of $K$ at $\mathfrak{p}$) be such that

where ${\mathfrak{p}}^m\parallel p$ and $r(p-1)>m$. Then there exists an $\alpha \in {\mathcal{O}}_{\mathfrak{p}}^{\times }$ such that ${\alpha }^p=\xi$.

Let $F=\mathbb{Q}({\zeta }_p)$ with ${\zeta }_p$ a primitive $p$-th root of unity. If $\xi \in {\mathcal{O}}_F$ and coprime to ${\lambda }_p:=1-{\zeta }_p$ and if

for some ${\alpha }_0\in {\mathcal{O}}_{F_{\mathfrak{p}}}^{\times }$ (the ring of integers of the completion of $F$ at ${\lambda }_p$), then the ideal $({\lambda }_p)$ is unramified in $K/F$ where $K=F(\sqrt[p]{\xi })$.

Let $p$ be a prime, $K=\mathbb{Q}({\zeta }_p)$ $\alpha \in K$ such that there exists $n\in \mathbb{N}$ such that ${\alpha }^n=1$, then $\alpha =\pm {\zeta }_p^k$ for some $k$.

Let $\alpha$ be an algebraic integer all of whose conjugates have absolute value one. Then $\alpha$ is a root of unity.

Let $K$ be a number field, $\alpha \in K$ and let ${\sigma }_i$ be the embeddings of $K$ into $\mathbb{C}$. Then

.

Let $K$ be a number field, $\alpha \in K$ and let ${\sigma }_i$ be the embeddings of $K$ into $\mathbb{C}$. Then

For $n$ any integer, ${\mathrm{\Phi }}_n$ (the $n$-th cyclotomic polynomial) is a polynomial of degree $\phi (n)$ (where $\phi$ is Euler’s Totient function).

For $n$ any integer, ${\mathrm{\Phi }}_n$ (the $n$-th cyclotomic polynomial) is an irreducible polynomial .

Let $f$ be a monic irreducible polynomial over a number field $K$ and let $\alpha$ be one of its roots in $\mathbb{C}$. Then

where the product is over the roots of $f$ different from $\alpha$.

Let $K=\mathbb{Q}(\alpha )$ be a number field, where $\alpha$ is an algebraic integer. Let $B=\{1,\alpha ,\dots ,{\alpha }^{[K:\mathbb{Q}]-1}\}$ be the basis given by $\alpha$ and let $x\in {\mathcal{O}}_K$. Then $\mathrm{\Delta }(B)x\in \mathbb{Z}[\alpha ]$.

Let $K$ be a number field and $B=\{1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{n-1}\}$ for some $\alpha \in K$. Then

where ${\sigma }_i$ are the embeddings of $K$ into $\mathbb{C}$. Here $\mathrm{\Delta }(B)$ denotes the discriminant.

Let $K$ be a number field with basis $B=\{b_1,\dots ,b_n\}$ and let ${\sigma }_1,\dots ,{\sigma }_n$ be the embeddings of $K$ into $\mathbb{C}$. Now let $M$ be the matrix

Then

Let $K=\mathbb{Q}(\alpha )$ be a number field, where $\alpha$ is an algebraic integer with minimal polynomial that is Eisenstein at $p$. Let $x\in {\mathcal{O}}_K$ such that $p^{n}x\in \mathbb{Z}[\alpha ]$ for some $n$. Then $x\in \mathbb{Z}[\alpha ]$.

Let $R$ be a Dedekind domain, $p$ a prime and $\mathfrak{a},\mathfrak{b},\mathfrak{c}$ ideals such that

and suppose $\mathfrak{a},\mathfrak{b}$ are coprime. Then there exist ideals $\mathfrak{e},\mathfrak{d}$ such that

Let $K$ be a number field and $B=\{b_1,\dots ,b_n\}$ be elements in ${\mathcal{O}}_K$, then $\mathrm{\Delta }(B)\in \mathbb{Z}$.

Let $p\ge 3$ be an prime number, ${\zeta }_p$ a $p$-th root of unity and $K=\mathbb{Q}({\zeta }_p)$. Assume that we have $x,y,z\in \mathbb{Z}$ with $gcd(xyz,p)=1$ and such that

This is the so called "case I". To prove Fermat’s last theorem, we may assume that:

• $p\ge 5$;

• $x,y,z$ are pairwise coprime;

• $x\not\equiv y\mathrm{mod}p$.

If $K$ is a number field and $\alpha \in {\mathcal{O}}_K$ then $N_{K/\mathbb{Q}}(\alpha )$ is in $\mathbb{Z}$.

Let $K=\mathbb{Q}(\alpha )$ be a number field with $n=[K:\mathbb{Q}]$ and let $B=\{1,\alpha ,{\alpha }^2,\dots ,{\alpha }^{n-1}\}$. Then

where $m_{\alpha }^{\prime }$ is the derivative of $m_{\alpha }(x)$ (which we recall denotes the minimal polynomial of $\alpha$).

If $K$ is a number field and $\alpha \in {\mathcal{O}}_K$ then ${\mathrm{Tr}}_{K/\mathbb{Q}}(\alpha )$ is in $\mathbb{Z}$.

Any unit $u$ in $\mathbb{Z}[{\zeta }_p]$ can be written in the form $\beta {\zeta }_p^k$ with $k$ an integer and $\beta \in \mathbb{R}$.

Let $p$ be an odd prime, ${\zeta }_p$ a primitive $p$-th root of unity and let $K=\mathbb{Q}({\zeta }_p)$. Then for any $i,j\in 0,\dots ,p-1$ with $i\ne j$, there exists a unit $u\in {\mathcal{O}}_K^{\times }$ such that ${\zeta }_p^i-{\zeta }_p^j=u({\zeta }_p-1)$.

Let $p$ be an odd regular prime. Then

has no solutions with $x,y,z\in \mathbb{Z}$ and $gcd(xyz,p)=1$.

Let $p$ be an odd regular prime. Then

has no solutions with $x,y,z\in \mathbb{Z}$ and $p|xyz$.

Let ${\zeta }_p$ be a $p$-th root of unity for $p$ an odd prime, let ${\lambda }_p=1-{\zeta }_p$ and $K=\mathbb{Q}({\zeta }_p)$. Then ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_p]=\mathbb{Z}[{\lambda }_p]$.

Let $p$ be a regular odd prime, $ϵ\in \mathbb{Z}[{\zeta }_p{]}^{\times }$ and $n\in {\mathbb{Z}}_{\ge 1}$. Then the equation $x^p+y^p+ϵ(1-{\zeta }_p{)}^{pn}{z}^p=0$ has no solutions with $x,y,z\in \mathbb{Z}[{\zeta }_p]$, all non-zero and $xyz$ coprime to $(1-{\zeta }_p)$.

Let $p$ be a regular prime and let $u\in \mathbb{Z}[{\zeta }_p{]}^{\times }$. If $u\equiv a\mathrm{mod}p$ for some $a\in \mathbb{Z}$, then there exists $v\in \mathbb{Z}[{\zeta }_p{]}^{\times }$ such that $u=v^p$.