1
vote
1answer
71 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
1
vote
1answer
34 views

Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the ...
2
votes
2answers
112 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
2
votes
2answers
54 views

Checking whether $\sqrt{2}$ is contained in the ideal $(2,\sqrt{-5}+1)$

Let $L=\mathbb{Q}(\sqrt{-5},\sqrt{2})$ and $I=(2,\sqrt{-5}+1)$ the ideal in $\mathcal{O}_L$, the ring of algebraic integers in $L$, which is generated by $2$ and $\sqrt{-5}+1$ I want to show that ...
3
votes
1answer
48 views

Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
1
vote
0answers
51 views

simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
2
votes
1answer
75 views

factorization of ideals

Let $L$ and $K$ be number fields such that $L/K$ is a finite extension. Suppose $\mathfrak{a},\mathfrak{b}$ are ideals in $\mathcal{O}_K$ and $\mathfrak{a}\mathcal{O}_L|\mathfrak{b}\mathcal{O}_L$. ...
4
votes
2answers
50 views

Special basis of an ideal as a $\mathbb{Z}$-module in number fields

I was speculating that the following may be true (but do not see any easy way to settle it; it must be known, I suppose): Given a (say, prime) ideal $\mathfrak{p}$ of the ring of integers ...
3
votes
3answers
96 views

Number of elements in $D/P^e$ where $D$ is a ring of algebraic integers, and $P$ a prime ideal

This is from Ireland and Rosen's A Classical Introduction to Modern Number Theory. Proposition 12.3.2: Consider a field $F/\mathbb Q$ with ring of integers $D$, and a prime ideal $P$ of $D$. Then ...
2
votes
3answers
83 views

Ideals in a real/complex number field?

Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$. Quick web search gave no ...
0
votes
0answers
49 views

Ideals in cubic fields

I've been studying number fields, and the ideals of their integer rings, and I have a question. First, I know the following in the quadratic case. If a $\mathbb{Z}$-basis for the integer ring is ...
2
votes
2answers
641 views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
15
votes
0answers
239 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
3
votes
1answer
43 views

Augmenting «$\Bbb Z[x]$ f.g. $\Rightarrow x$ integral» for ${\frak p}[x]$

In KCd's blurb on ideal factorization, page 5: $\hskip 0.3in$ The situation is this: $K$ is a number field, ${\cal O}_K$ its ring of integers, ${\frak p}\triangleleft{\cal O}_K$ a prime ideal, ...
2
votes
2answers
80 views

Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
1
vote
1answer
104 views

Showing Quotient Ring is a Field

Consider the ring $S=\mathbb{Z}[\alpha]$, where $\alpha = \sqrt[3]{2}$, and ideal $I=(5,\alpha^{2}+3\alpha -1)$. I wish to show that $S/I$ is a field of order 25. Any solutions/suggestions? I would ...
2
votes
0answers
66 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
1
vote
1answer
163 views

How do we find prime ideals of a ring of integers of a number fileld?

For example for $F=Q(\sqrt{-5})$. the ring of integers of $F =Z[\sqrt{-5}]$.(since $-5\equiv3 \pmod 4$) but how can we determine prime ideals of this? and another problem is the corresponding ...
2
votes
1answer
141 views

How to calculate the norm of an ideal?

Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly. For example, I know that the norm ...
11
votes
3answers
310 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
3
votes
2answers
255 views

Is the inverse of a fractional ideal still fractional?

Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define ...
4
votes
3answers
351 views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
4
votes
1answer
56 views

Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
3
votes
1answer
166 views

Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
4
votes
2answers
88 views

Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
3
votes
2answers
122 views

Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
4
votes
1answer
149 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
6
votes
1answer
98 views

Factorization of $5$ in the splitting field of $x^3 + 2$

I wonder if someone could help to clarify the following. Let $\zeta$ be a primitive cube root of unity and $\alpha = \sqrt[3]{2}$. Let $K = \mathbb{Q}(\alpha)$ and $L = K(\zeta)$. Then $L$ is the ...
3
votes
3answers
225 views

Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
1
vote
1answer
81 views

Fractional ideals in quadratic field extension

I have some problems with the topic "fractional ideals". I have two questions: Compute a generator $\alpha$ of the fractional ideal $\Bbb{Z}+\Bbb{Z}(\phi^3(5+\sqrt{31}))$, thus find ...
7
votes
3answers
300 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
0
votes
2answers
49 views

Why ideal containment in the proof of unique ideal factorization?

I want to understand the proof that the number fields (which are dedekind domains) have unique factorization of ideals. I am trying to read this proof here IDEAL FACTORIZATION - KEITH CONRAD but.. ...
3
votes
1answer
85 views

Equivalence of Valuations - Trouble Understanding Proof

I want to complete the proof of the following theorem. Here is what I have got so far: Theorem Every non-euclidean valuation $v$ on a number field $K$ is equivalent to $v_{\mathfrak p}$ for some ...
3
votes
1answer
280 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
11
votes
5answers
474 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
6
votes
1answer
397 views

How to show this ideal is not principal

I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and ...
5
votes
1answer
118 views

Three maximal ideals lying over $3\mathbb{Z}$?

A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
4
votes
1answer
481 views

Are distinct prime ideals in a ring always coprime? If not, then when are they?

Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when ...
2
votes
2answers
88 views

Can one define “addition” of ideals to correspond to addition of numbers?

(In the setting of number fields and algebraic integers) If $(a),(b)$ are two principle ideals then $(a)+(b)$ corresponds to $(\gcd(a,b))$, so while the natural definition of addition for ideals has a ...
8
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
1
vote
0answers
136 views

Prime ideal splitting in field extension and its normal closure

The question is: Let L / K be a finite (not necessarily Galois) extension of algebraic number fields and N / K the normal closure of L / K. Show that a prime ideal p of K is totally split in L if and ...