Questions related to the algebraic structure of algebraic integers

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5
votes
1answer
31 views

Subfield of a cyclotomic number field where a prime $p$ is inert

I am reading this paper by Adleman,Lenstra on finding Irreducible polynomial over Finite field. Here in Section VI(Proof of correctness of Algo B) I came across this argument: Let $q_i $ be a prime ...
1
vote
1answer
44 views

A prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ lies above/over $p$ if $\mathfrak{p}\cap \mathbb{Z} = p\mathbb{Z}$ [duplicate]

In the concrete case that $\mathcal{O}_K = \mathbb{Z}[i]$, and $\mathfrak{p} = (1+i)$, how to make sense of $\mathfrak{p}\cap \mathbb{Z}$? I want to know if (1+i) lies above/over 2.
6
votes
1answer
73 views

Finding the ring of integers of $\Bbb Q(\sqrt[4]{2})$

I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt[4]{2})$ is $\Bbb Z[\sqrt[4]{2}]$ and I would like to prove it. A related question is this one, but it doesn't answer mine. I computed ...
4
votes
1answer
49 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
4
votes
1answer
40 views

Example of field's normal closure that's not Abelian?

Suppose $K$ is a global field, $L/K$ is a field extension, and $M$ = normal closure of $L$ (over $K$). Is it possible that Gal($M/L$) is not Abelian? In all cases I know, $L$ is formed from $K$ by ...
2
votes
1answer
62 views

what is the value of $\zeta_{\mathbb{Q}(i)}(-1)$?

We've been told over and over $\boxed{\zeta(-1) = 1 + 2 +3 + 4 + \dots = - \frac{1}{2}}$ can be do the same over number fields? What should be the reasonable value for the zeta function $F = \mathbb{...
3
votes
2answers
31 views

Determining whether an element belongs to ring of integers

Consider a number field $K=\mathbb{Q}[\alpha]$ and we wish to show that some other element $\beta \in K$ belongs to the ring of integers $\mathcal{O}_K$. Is it enough to show that the norm and trace ...
1
vote
1answer
27 views

Discriminant of real cyclotomic field

I know following theorem (and its proof): Let $K\subset L \subset M$ be number fields, $[L:K] = n, [M:L]=m$, and let $\{\alpha_1,\ldots,\alpha_n\}$ and $\{\beta_1,\ldots,\beta_m\}$ be bases for $L$...
3
votes
1answer
40 views

Valuation of Index of polynomial with Newton Polygon

I read here (page 237) that the valuation of the index of a polynomial is equal to the number of integer points below its Newton polygon. I am confused how this makes sense--the cited paper (this) ...
2
votes
2answers
45 views

Example of a number field with only one complex place

In a number theory textbook they are asking for a field $F$ which has only one complex place. Can $F = \mathbb{Q}(\sqrt{-2})$ ? Can $F$ be or arbitrary degree?
1
vote
1answer
46 views

What is meant by a number to be a root of unity?

I am proving first case of Fermat's last theorem for regular primes by following Marcus' book "Number Fields". I have to prove following statement: If $\varepsilon$ is a unit in $\mathbb{Z}[\omega]...
1
vote
1answer
36 views

Completion and Algebraic Closure

Suppose we start with a valued field $K$ and we want to find a field extension of $K$ that is algebraically closed and complete. The usual process is: Consider the completion $\hat{K}$ of $K$, then ...
0
votes
1answer
41 views
+100

Adeles for function fields

Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is: $$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$ where $v$ ranges ...
4
votes
2answers
73 views

$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
2
votes
2answers
35 views

Embeddings of pure cubic field in complex field

I know that the complex embeddings (purely real included) for quadratic field $\mathbb{Q}[\sqrt{m}]$ where $m$ is square free integer, are $a+b\sqrt{m} \mapsto a+b\sqrt{m}$ $a+b\sqrt{m} \mapsto a-b\...
3
votes
1answer
80 views

Fraction field of $p$-adic power series ring

Let $L$ be a finite extension of $\mathbf{Q}_p$. Write $$\mathcal{O}_{\mathcal{E}} = \left\{ f = \sum_{k \in \mathbf{Z}} a_kT^k \in \mathcal{O}_{L}[[T,T^{-1}]] \mid \lim_{k \to -\infty} a_k = 0\right\...
11
votes
0answers
69 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
1
vote
1answer
28 views

What is the definition for totally ramified extension for a global field?

What is the definition for totally ramified extension for a global field? For local fields it means the maximal prime ideal generated from the uniformizer totally ramifies. But what is the definition ...
1
vote
1answer
60 views

Have I found an example of norm-Euclidean failure in $\mathbb Z [\sqrt{14}]$?

Based on the proof that $\mathcal O_{\mathbb Q (\sqrt{-19})}$ is not Euclidean because it lacks universal side divisors, I have convinced myself that $\mathbb Z [\sqrt{14}]$ is Euclidean because it ...
1
vote
0answers
28 views

Completion of abelian extension

I don't understand this step in a proof: Let $K/\mathbb{Q}$ be an abelian extension, $p$ a ramified prime and $K_p$ the completion at that prime. Then $K_p/\mathbb{Q}_p$ is abelian. Why does ...
2
votes
0answers
33 views

factoring polynomials in ring of integers modulo powerful number

I am having trouble finding info on how to factor polynomials in ring of integers modulo powerful number. For example: $x^2 - 1$ in $\textbf Z_{8}$. I know by tinkering around that $(x - 1)(x + 1)$...
4
votes
2answers
37 views

Ring of algebraic integers as lattice points in the complex plane

Let, $i=\sqrt{-1}$ and $\omega = e^{\frac{2\pi i}{3}}$. I know that we can represent the ring of integers $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$ as square and triangular lattice on complex plane ...
5
votes
2answers
94 views

If $f$ has more than one root in $K$, then $f$ splits and $K/k$ is Galois?

Let $f \in k[x]$ be an irreducible polynomial of prime degree $p$ such that $K \cong k[x]/f(x)$ is a separable extension. How do I see that if $f$ has more than one root in $K$, then $f$ splits and $K/...
1
vote
1answer
19 views

Normal closure of a number field and a possible quadratic field in it

While reading about prime decomposition in number fields, I came across following statement (stated as a fact): Let $K$ be a number field and $d= \text{disc}(\mathcal{O}_K)$, then the normal ...
1
vote
1answer
41 views

Splitting of primes in real cyclotomic field

The question is from Marcus' book, "Number Fields" (exercise 12, Chapter 4) Let $\omega= e^{\frac{2\pi i}{m}}$ and $p$ be a rational prime not dividing $m$. Then how does $p$ split in $\mathbb{...
3
votes
1answer
28 views

Factorization of primes in normal closure of Quartic Field

Motivation for the question comes from Marcus' book on Number Fields (exercise 13, Chapter 4). Let $K= \mathbb{Q}[\sqrt[4]{m}, i]$ where $i=\sqrt{-1}$, $m\in \mathbb{Z}$ and $m$ is not a square. ...
0
votes
2answers
77 views

Find remainder when $f(x^{12})$ is divided by $f(x)$

What will be the remainder when $f(x^{12})$ is divided by $f(x)$ where : $$f(x) = x^5 + x^4 + x^3 + x^2 + x +1$$ I have already tried but found no idea how I can do this question.
18
votes
3answers
348 views

A few questions on the Gaussian integers

I have a few questions surrounding the Gaussian integers, which I hope can be answered together in one fell swoop. The Gaussian integers are defined as $\mathbb{Z}[i] = \{x + iy : x, y \in \mathbb{Z}...
0
votes
2answers
19 views

is the compositum of a family of algebraic extensions algebraic?

Let $F$ be a field contained inside another field $K$. Let $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of algebraic extensions of $F$ ...
4
votes
2answers
93 views

Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
3
votes
2answers
54 views

Sum of squares using Gaussian integers

Using Gaussian integers $a+ib$ ($a,b\in\mathbb{Z}$), one can prove that a prime $p\in\mathbb{Z}$ is sum of two squares in $\mathbb{Z}$ if and only if $p\equiv 1\pmod 4$. Question: Using Gaussian ...
2
votes
0answers
35 views

Does every coset in a ring of integers contain a totally positive element?

Let $K$ be a number field with ring of integers $\mathcal O_K$, let $\mathfrak m$ be an ideal in $\mathcal O_K$ and let $a \in \mathcal O_K$ such that $(a, \mathfrak m) = 1$. Does there necessarily ...
1
vote
1answer
54 views

Is there a redundant assumption in Exercise 2, page 14, from Janusz “Algebraic Number Fields”?

The exercise says: Let $R$ be an integral domain with quotient field $K$ and let $M$ be an $R$-submodule of a finite dimensional $K$-vector space. Prove $M=\bigcap_{P} R_P M$, where the ...
3
votes
0answers
37 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
5
votes
2answers
128 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \...
1
vote
2answers
65 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find the square root for the Number Field Sieve (reference here page 4 first line). It says that for this method the degree of the extension $K/\mathbf Q$ must ...
2
votes
1answer
35 views

Understanding the notation of a paper

I am reading a paper on Algebraic Number Theory that says If $p$ divides the discriminant of polynomial $f$ $r$ times and there is the factorization into irreducibles $$f(x)\equiv g_1(x)\dots g_r(...
3
votes
0answers
74 views

A prime number $p$ is ramified in $\mathbb{Q}(\sqrt[p]{a})$.

Let $p$ be an odd prime number and $a\in \mathbb {Z}$ with $\sqrt[p]{a}\notin \mathbb{Z} $. Prove that $p$ is ramified in the number field $\mathbb{Q}(\sqrt[p]{a})$. My idea is to apply Dedekind's ...
2
votes
1answer
32 views

Existence of units in number fields outside the rings of integers

Let $K/ \mathbb Q$ be a number field with $[K:\mathbb Q]=n$. Using that there exists a prime $p\in \mathbb Z$ which splits completely, that is $p\mathcal O_K=P_1...P_n$ for some distinct primes $P_i$ ...
0
votes
1answer
27 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
1
vote
0answers
50 views

Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
2
votes
0answers
38 views

Trace of Witt vectors

Let $\mathbb{F}_p$ be a finite field with $p$ elements, and $\kappa := \mathbb{F}_q$ an extension of $\mathbb{F}_p$ of degree $n$ with $q = p^n$. Then $W_\infty(\kappa)$ is a ring extension of $W_\...
2
votes
0answers
75 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
6
votes
1answer
57 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...
5
votes
1answer
104 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
2
votes
0answers
37 views

If $p$ is unramified in every subfield of $K$, does it mean $p$ is unramified in $K$?

I am wondering if $p$ being unramified in every subfield of $K$ means $p$ is unramified in $K$. Any hints?
3
votes
1answer
61 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
5
votes
1answer
42 views

When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
0
votes
1answer
31 views

Class group embedding in coprime extension

Let $L/K$ be an extension of number fields of degree $n$. Assume that the class group of $K$ has order $h$. Prove that if $(h,n)=1$ the map $Cl(K)\rightarrow Cl(L)$, given by $I\rightarrow I\mathcal ...
1
vote
2answers
43 views

Reference request: Binary quadratic forms

I am currently a first year grad student doing an independent study on topics in algebraic number theory and am currently looking at some of the properties of the polynomial $n^2 + n + A$, where $A \...