Questions related to the algebraic structure of algebraic integers

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-3
votes
3answers
244 views

Norm of a fractional ideal of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank $n$. The ideal theory of $R$ is useful at least when ...
-3
votes
1answer
131 views

Determination of the prime ideals lying over an odd prime in a quadratic order

We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of ...
-4
votes
1answer
155 views

Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
-4
votes
3answers
42 views

If α is algebraic over K then all the elements of K(α) are algebraic over K [on hold]

Let $L$ : $K$ be a field extension, $\alpha \in L$ algebraic over $K$. Show that every element of $K(\alpha)$ is algebraic over $K$, where $K(\alpha)$ is the smallest subfield that contains $\alpha$ ...
-5
votes
2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
-5
votes
1answer
269 views

Real units of the cyclotomic number fields of an odd prime order

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\epsilon$ be a unit ...
-6
votes
1answer
224 views

On equivalence of primitive ideals of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$. Let $D$ be the discriminant of $R$. Let $x_1,\cdots, x_n$ be a ...