2
votes
2answers
39 views

Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
2
votes
1answer
35 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
3
votes
1answer
25 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
2
votes
1answer
35 views

Ideals in a Quadratic Number Field

Show the ideal $I=\langle4,2+2\sqrt{-29}\rangle$ in $\mathbb{Z}[\sqrt{-29}]$ satisfies the equality $\langle8\rangle=I^{2}$ of ideals in $\mathbb{Z}[\sqrt{-29}]$. I tried to factorise $x^{2}+29$ over ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
2
votes
0answers
31 views

Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
0
votes
0answers
25 views

Ireland and Rosen, question 12.28 (unique factorization in prime ideals)

I'm stuck with the following exercise in Ireland and Rosen, chapter 12. Let $D$ be the ring of integers in a number field $F$. Suppose $(p)=P^2A$ for $p$ prime in $\Bbb Z$ and a prime ideal $P$. ...
1
vote
1answer
46 views

Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
0
votes
1answer
21 views

multiplying ideals

At the moment I am learning algebraic number theory and I have seemed to be missing some basic understanding. How do you multiply ideals For example $$(2,1+\sqrt{-5})^2 = ...
1
vote
1answer
81 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
43 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
140 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
61 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
59 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
55 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
86 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
65 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
103 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
99 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
54 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
780 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. ...
18
votes
1answer
313 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
46 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
81 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
115 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
69 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
254 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
264 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
340 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
288 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
446 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
64 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 ...
4
votes
1answer
215 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
92 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
144 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
168 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
109 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 ...
2
votes
3answers
238 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
85 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 ...
8
votes
3answers
333 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
53 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
94 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 ...
4
votes
2answers
342 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 $\mathbb{Q}(\sqrt -2).$ ...
12
votes
5answers
520 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
415 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
119 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
515 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
91 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 ...
9
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
142 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 ...