# Tagged Questions

0answers
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### Prove that $\mathfrak{p}$ is totally split in $L/K$ and $L'/K$ $\Rightarrow$ totaly split in $LL'/K$

Assume that $K$ be a number field and $L/K$, $L'/K$ are two separable extensions. Now let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$. Then if $\mathfrak{p}$ is totally split ind $L$ and $L'$, ...
1answer
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### An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
2answers
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### 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 ...
1answer
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### 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 $\mathcal{O}_K$. ...
1answer
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### 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 ...
1answer
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### 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 ...
2answers
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 ...
1answer
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5answers
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### 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 ...
1answer
422 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 ...
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 ...
1answer
522 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 ...
2answers
92 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 ...
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 ...