Dependencies

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 \(K\) be a number field and \(B,B'\) bases for \(K/\mathbb {Q}\). If \(P\) denotes the change of basis matrix, then

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

For \(i \neq j\) 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 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 \(p \geq 5\) be an prime number, \(\zeta _p\) a \(p\)-th root of unity and \( \alpha \in \mathbb {Z}[\zeta _p]\). If there is an integer \(n\) such that \(\alpha /n \in \mathbb {Z}[\zeta _p]\), then \(n\) divides each \(a_i\).

Let \(p \geq 5\) be an prime number, \(\zeta _p\) a \(p\)-th root of unity and \( \alpha \in \mathbb {Z}[\zeta _p]\). Suppose 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 \(K\) be a number field and let \(B=\{ b_1,\dots ,b_n\} \) be a set of elements in \(K\). Then \(\Delta (B) \neq 0\) if and only if the elements in \(B\) are linearly independent.

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

For \(n\) any integer, \(\Phi _n\) (the \(n\)-th cyclotomic polynomial) is an irreducible polynomial of degree \(\varphi (n)\) (where \(\varphi \) is Euler’s Totient function).

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 \(K\) be a number field and \(B,B'\) bases for \(K/\mathbb {Q}\). If \(P\) denotes the change of basis matrix, then

Let \(K\) be a number field and \(B=\{ b_1,\dots ,b_n\} \) be a basis for \(K/\mathbb {Q}\) consisting of algebraic integers. If \(B\) is not an integral basis then there exists an algebraic integer of the form

where \(p\) is a prime and \(x_i \in \{ 0,\dots ,p-1\} \) with not all \(x_i\) zero. Moreover, if \(x_i \neq 0\) and we let \(B'\) be the basis obtained by replacing \(b_i\) with \(\alpha \), then

In particular \(p^2 \mid \Delta (B)\).

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 \(\Delta (B)\) 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 \(K=\mathbb {Q}(\alpha )\) and \(\alpha \) be an algebraic integer such that \(m_\alpha \) satisfies Eisensteins Criterion for a prime \(p\). Then none of the elements

is an algebraic integer, where \(n=\deg (m_\alpha )\) and \(x_i \in \{ 0,\dots ,p-1\} .\)

Let \(p\) be a prime and \(n=p^k\). Then

where \(u \in \mathbb {Z}[\zeta _n]^{\times }\).

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 \(\Delta (B) \in \mathbb {Z}\).

Let \(p \geq 5\) 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

Then without loss of generality, we may assume \(x,y,z\) are pairwise coprime and

If \(K\) is a number field and \(\alpha \in \mathcal{O}_K\) then \(\operatorname {Tr}_{K/\mathbb {Q}}(\alpha )\) and \(N_{K/\mathbb {Q}}(\alpha )\) are both 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 \(m_\alpha '\) is the derivative of \(m_\alpha (x)\) (which we recall denotes the minimal polynomial of \(\alpha \)).

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 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 \(\mathcal{O}_K=\mathbb {Z}[\zeta _p]=\mathbb {Z}[\lambda _p]\) moreover

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