# Tagged Questions

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### $\mathbb{Q}(i)$ has no unramified extensions

It is a classical result that every extension of $\mathbb{Q}$ is ramified. Put differently: there are no unramified extensions of $\mathbb{Q}$. The classical proof follows from the following two ...
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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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### Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
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### Genus field = Hilbert Class Field (Cox exercise 6.15)

Prove that the genus field of an imaginary quadratic field of $K$ equals its Hilbert Class Field if and only if for primitive forms of discriminant $d_k$, there is only one class per genus. ...
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### What is the proof Shafarevich gave that solvable groups are realizable over $\mathbb{Q}$?

I want to study it and present in front of my faculty as my seminar. I can't find the proof online anywhere. One proof I found was constructive and was for global fields. I want it over $\mathbb{Q}$ ...
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### $Tr(\alpha_{i}^{n})=\alpha_{1}^{n}+…+\alpha_{d}^{n}$

I want to show $Tr(\alpha_{i}^{n})=\alpha_{1}^{n}+...+\alpha_{d}^{n}$ where $\alpha_{i}$ is algebraic integer of degree d and the $\alpha_{j}$ are it's conjugates. Is there any quick way to show it ...
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### Integral Closure in an Unramified Extension is Generated by a Single Element

Let $R$ be a discrete valuation ring with quotient field $K$, and $L/K$ a finite separable extension which is unramified over $K$. Also suppose that $K$ is complete with respect to the valuation of ...
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### Using discriminants to find order of extension

Any hints on how to show $[G:H]^{2}=\frac{disc(H)}{disc(G)}$,where G,H are free abelian groups of rank n and $H\subset G\subset K$,where K is a number field? Alternative formulation, how to relate ...
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### The basis for additive subgroup is discriminant-invariant

i.e. given bases $\{\beta_{i}\}$ and $\{\gamma_{i}\}$ for S additive subgroup of number field K (degree n over $\mathbb{Q}$), then $disc(\{\beta_{i}\})=disc(\{\gamma_{i}\})$. any hints? Is that ...
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### Basis for $O_{Q[\alpha]}$ existence theorem reference

Theorem 13 from Marcus Number fields "This is an integral basis $1,\frac{f_{1}(\alpha)}{d_{1}},\frac{f_{2}(\alpha)}{d_{2}},...,\frac{f_{n-1}(\alpha)}{d_{n-1}}$ ,where $f_{i}$ monic and ...
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### Which remarkable properties does the Hilbert Class Field have?

Let $L$ be the Hilbert Class Field of $K$, then: $Gal(L/K) \cong CL(K)$ by Artin reciprocity. though being Galois is not transitive in general, we nonetheless have for $K/\mathbb{Q}$ and $L/K$ ...
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### 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 ...
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### Ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$

Let $p,q$ be distinct odd primes, and $\omega_p,\omega_q$ primitive $p$-th, $q$-th roots of unity. What is the ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$? The numbers in the ring ...
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### In which extensions of type $\mathbb{Q}(\sqrt{d})$ is the number 3 reducible?

Let $O$ be the ring of integers in $\mathbb Q(\sqrt{d})$ where $d$ is a square free negative number. Assume that the number 3 is reducible in $O$. Prove that $d = -2,-3,$ or $-11$. Thank you for the ...
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### $P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $\Phi_{\mathfrak m,L|K}$ and $\Phi_{\mathfrak m,M|K}$. Let ...
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### Field of all algebraic reals over $\mathbb{Q}$ has infinite order.

I am trying to show that field of all algebraic reals over $\mathbb{Q}$ has infinite degree. I guess that $$1,\sqrt{2},\sqrt[3]{2}, \sqrt[4]{2}, ...$$ are lineary independent but can't prove it.
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### Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
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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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### 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 $\mathfrak{a}\mathcal{O}_L|\mathfrak{b}\mathcal{O}_L$. ...
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### An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
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### $p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
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### Questions about a proof in Greenberg's Book.

I am trying to understand the proof of the following lemma : Lemma ' : Suppose that $X$ is a finitely generated $\Lambda$-module ($\Lambda =\mathbb Z_p[[T]]$) and that ...
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### unique factorisation fails for cyclotomic integers $p>23$

Background: I have stopped doing algebra a long time ago and I am not that interested in the nitty-gritty details of proofs, but I am interested in maths history. ...
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### cubic extension, ramfication [closed]

Let $L = K(\sqrt[3]{m})$ be a cubic extension of $K$ where $m$ is a cubefree integer and $K$ is an imaginary quadratic field. If $\mathfrak{p}$ is any prime of $K$, dividing $m$, then prove that ...
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### Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
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### Unique factorization of Ideals?

Is it case that even if the domain is not UFD for its elements, the domain is UFD for ideals. I mean can we uniquely factorized the ideals, whatsoever? possible, and why? for example, in ...
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### Give an example of a UFD having a subring which is not a UFD.

Give an example of a UFD having a subring which is not a UFD. I thought of $\mathbb{Z}[\sqrt{2},\sqrt{3}]$. Could you please explain my question. I am trying grasp the concepts, need help.
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### Find the ideal class group of $\mathbb{Z}(\frac{1+\sqrt{-31}}{2})$.

Find the ideal class group of $\mathbb{Z}(\frac{1+\sqrt{-31}}{2})$. I found the the ideal class group(ICG) is generated by primes ideals lying over 2 and 3. $\lt 2 \gt$ =$P_2 \hat P_2$ $\lt 3 \gt$ ...
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### $Q_p(\zeta)$ where $\zeta$ is a $p$-th root of $1$.

I'm not looking for a full solution, only a hint please! Let $\zeta$ be a $p$-th root of unity in an algebraic closure of $Q_p$. Show that $Q_p(\zeta) = Q_p ((-p)^{\frac{1}{p-1}})$. Following a hint ...
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### In a Dedekind domain every ideal is either principal or generated by two elements.

Prove that in a Dedekind domain every ideal is either principal or generated by two elements. Help me some hints. Thanks a lot.
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### Lattice Bases of Prime (Ideal) Divisors

My question is: How can I find the prime (ideal) divisors of 2 and 3 in the ring of integers of $\mathbb Q[\sqrt{-14}]$ and $\mathbb Q[\sqrt{-23}]$? Here's what I have so far. I found that (2, ...
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### Ideals in $\mathbb Z [\sqrt {-6}]$ are principal?

Are the ideals in $\mathbb Z [\sqrt {-6}]$ principal? I know from Artin that there is a method of trying to cover the fundamental parallelogram of $\mathbb Z [\sqrt {-6}]$ with disks, but I am ...
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### 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 ...
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### 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 ...
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### Algorithm for finding a basis of a subgroup of a finitely generated free abelian group

Let $G$ be a finitely generated free abelian group. Let $\omega_1,\cdots, \omega_n$ be its basis. Suppose we are given explicitly a finite sequence of elements $\alpha_1,\cdots, \alpha_m$ of $G$ in ...
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### Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$?

This a step in my notes which I can't seem to understand clearly. Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$? I see ...
### 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. ...
How to show $$5\mathbb{Z}[\sqrt[3]{2}] = (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1).$$ Firstly, I think that I can say that (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1)= ...