# Tagged Questions

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### Theory of irrationalities- Faddeev's book

Does anyone know where (if available) I can get a free access to Delone, B. N., Faddeev, D. K., ''The theory of irrationalities of the third degree'' Transl. Math. Monographs 10, Amer. Math. Soc., ...
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### integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
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### 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 ...
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### Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...
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### Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
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### The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is ...
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### Determining ring of integers for $\mathbb{Q}[\sqrt{17}]$

I'm trying to find the ring of integers of $\mathbb{Q}[\sqrt{17}]$, and it comes down to determining the set $\{(a,b)\in\mathbb{Q}^2\mid 2a\in \mathbb{Z}, a^2-17b^2\in\mathbb{Z}\}$. How can I ...
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### quadratic rings of integers vs cubic rings of integers in number fields

I would appreciate if someone could give me some clues about cubic $\mathbb{Z}$-rings of number fields. So far I have only learned about quadratic rings and I would like to see if there are any ...
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### Non-unique factorization of an ideal in UFD

In Z[x], the ideal <2, x> is not principal. I am that the factorization of a nontrivial ideal into prime ideals is unique in a Dedekind domain. Not all UFD are Dedekind domain, so there must be a ...
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### Examples of Dedekind rings with infinite class number

I am looking for explicit examples of Dedekind rings with infinite class number. In most books on algebraic number theory there is a standard example (before or after proving that the class number is ...
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### Irreducibility of a particular polynomial

I've got this problem for my homework: find out whether the polynomial $$f(x)=x(x-1)(x-2)(x-3)(x-4) - a$$ is irreducible over the rationals, where $a$ is integer which is congruent to $3$ modulo $5$. ...
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### 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. ...
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### Overrings of Dedekind domains as localizations

I am taking an independent study where I organize and present weekly material on algebraic number theory to my professor and receive feedback. Next week I am going to cover some miscellaneous topics, ...
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### A certain ideal of a valuation ring

This is a question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
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### 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 ...
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### The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
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### Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those ...
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### 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 ...
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### Ring of integers of a degree $5$ extension

Consider the polynomial $P(X) = X^5 - X + 1 \in \mathbb{Q}[X]$, and let $x \in \mathbb{C}$ be a root of $P(X)$. Let $K = \mathbb{Q}(x)$. How can you prove that the ring of integers $\mathcal{O}_K$ is ...
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### How to show $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$?

Let $\mathbb{Z}[w]=\mathbb{Z}[\frac{1+\sqrt{-15}}{2}]$ be the quadratic integers. I want to show that $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$. It seems very clear, but how can I show the ...
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### fraction field of the integral closure

Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of ...
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### Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
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### 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 ...
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### Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
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### Proving an ideal is principal

Let $R,\mathfrak{P},\overline{\mathfrak{P}}$ and $p$ be as in this question. I have proved that $\mathfrak{P}\cdot\overline{\mathfrak{P}}=pR$. I think this can be used for proving what follows, by I ...
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### Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
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### Integral domains such that all proper factor rings are finite

Let $\mathbb Z$ be the ring of rational integers. If $a\in\mathbb Z$ is a non-zero element, then the factor ring $\mathbb Z/(a)$ is finite and has order $|a|$. If $\mathbb Z[i]$ is the ring of ...
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### How many prime ideals does $\mathbb Q[x]/(x^m -1)$ have? (multiple choice)

Let $m$ be a positive integer, and $a_m$ denote number of distinct prime ideals of $\mathbb Q[x]/(x^m -1)$. Then which of the following are true? $a_4=2$ $a_4=3$ $a_5=2$ $a_5=3$
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### Norm of the generators of a fractional ideal.

Let $\mathcal{O}_l=\mathbb{Z}[\frac{1+\sqrt{-l}}{2}]$ with $l$ a prime number congruent to 3 mod 4. Let $\mathfrak{a}$ be a non-principal fractional ideal of $\mathcal{O}_l$. My questions are: Why ...
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### Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?

I know that $\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain under the usual norm $N(x + y\sqrt{-3}) = x^2 + 3y^2$, but that does not necessarily mean that it can't be a Euclidean domain. Is it ...
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### Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
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### Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$

I know that $\mathbb Z[\sqrt{7}]$ is a UFD, and I can write the equation $(2 + \sqrt{7})(3 - 2\sqrt{7}) = (5 - 2\sqrt{7})(18 + 7\sqrt{7})$. So clearly these are not all irreducibles. How do I ...
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### 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 ...
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### What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
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### Fermat's Last Theorem and Kummer's Objection

In 1847 LamÃ© had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
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### about the fractional ideal of a field of fractions

In the wikipedia article http://en.wikipedia.org/wiki/Fractional_ideal we read Let $R$ be an integral domain, and let $K$ be its field of fractions. A fractional ideal of $R$ is an $R$-submodule ...
Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field. For ...
Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...