Questions related to the algebraic structure of algebraic integers
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75 views
Quadratic Diophantic equation
Hello :) i want to give a answer op the following question:
For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$.
I want to solve the question without ...
5
votes
1answer
308 views
Prove that the equation $y^2=x^3-73$ has no integer solutions
Prove that there are no integers $x,y$ such that $y^2=x^3-73$.
Thank you.
5
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1answer
138 views
Two corollaries in Lang's Algebraic Number Theory.
I'm having difficulty understanding the relationship between two corollaries in Lang's Algebraic Number Theory, on page 16 for those with the book. They can also be found in his Algebra.
The first ...
5
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1answer
70 views
Quadratic field such that a certain finite set of primes split
Given a finite set $S$ of primes, is it possible to find an imaginary quadratic field $K$ such that all primes in $S$ are split completely in $K$?
5
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1answer
152 views
Extending Galois automorphism to group automorphism
Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every ...
5
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1answer
188 views
A Number Field that's Galois over $\mathbb{Q}$ is either totally real or totally imaginary
I came across the following assertion in the Wikipedia article about totally imaginary number fields.
Let $K/\mathbb{Q}$ be an algebraic number field that is Galois over $\mathbb{Q}$. Then ...
5
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1answer
531 views
Ramification index and inertia degree
Let $L,K$ be number fields and $L|K$ a galois extension.
Let $(0)\neq Q$ a prime ideal in $\mathcal O_L$ (=ring of integers in $L$) and $P=Q \cap \mathcal O_K$.
$Z_Q $ denotes the decomposition ...
5
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1answer
37 views
Extension of valuation to the algebraic extension of a number field.
I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to ...
5
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1answer
65 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 ...
5
votes
1answer
147 views
A Gauss sum like summation
I would like to calculate the following sum.
Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime.
The sum is
$$\sum_{j=1}^n (-1)^j ...
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1answer
77 views
Definition of tamely ramified
I think I can show that the following definitions of "tamely ramified" coincide. I thought it would be good to be sure. Sorry for the easy questions.
Let $O_K$ be a dvr with maximal ideal $\mathfrak ...
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1answer
119 views
Typo in Marcus' $\textit{Number Fields}$?
I am doing Problem 5.10 of Marcus where it is given that $m$ is a square-free negative integer and that $\mathcal{O}_K$ is a PID where $K = \Bbb{Q}(\sqrt{m})$. Now in part (b) of this problem he ...
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1answer
65 views
Equivalent Definition of Non-Archimedean Local Field
Wikipedia states that there is an equivalent definition of non-archimedean local fields: "it is a field that is complete with respect to a discrete valuation and whose residue field is finite." ...
5
votes
2answers
191 views
Integral Basis for Cubic Fields
I'm trying to follow a text (Lang's Algebraic Number Theory) in which it fully determines an integral basis for quadratic fields (also seen here). Is there any easy or analogous way to determine one ...
5
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1answer
171 views
Proving the class number of $\mathbb{Q}(\sqrt{-5})$ is 2 using Ireland-Rosen's bound
In this MO answer, Keith Conrad states that you can use the method of proof of the finiteness of the class number in Ireland & Rosen to prove that the class number $h_K=2$, when ...
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votes
3answers
303 views
On the ring generated by an algebraic integer over the ring of rational integers
Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial.
Let $\theta$ be a root of $f(X)$.
Let $A = \mathbb{Z}[\theta]$.
Let $p$ be a prime number.
Suppose $p$ does not divide the discriminant ...
5
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1answer
175 views
Is there an efficient algorithm to compute a minimal polynomial for the root of a polynomial with algebraic coefficients?
An algebraic number is defined as a root of a polynomial with rational coefficients.
It is known that every algebraic number $\alpha$ has a unique minimal polynomial, the monic polynomial with ...
5
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1answer
219 views
Algorithm for finding the discriminant of algebraic number fields
I am reading J.S. Milne's Algebraic Number Theory notes, http://jmilne.org/math/CourseNotes/ANT.pdf. I am quite confused with the section "Algorithm for finding the ring of integers".
There is an ...
5
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1answer
87 views
The conductor and minimal moduli of abelian extensions
Assume that $L/K$ is a finite abelian extension of global fields and $S$ the set of primes of $K$ ramifying in $L$. Then the conductor $\mathfrak{f}(L/K)$ is the smallest modulus s.t. the Artin map
...
5
votes
1answer
159 views
A generalization of Ostrowski's Theorem
Let $D$ be a dedekind domain and $|\cdot|$ be a bounded, nontrivial norm on $D.$ Is $|\cdot|$ equivalent to an $I$-adic norm for some maximal ideal $I$ of $D?$
5
votes
1answer
103 views
Alternative integral basis for $\Bbb{Z}[w]$
Write $w= e^{2\pi i/m}$ for $m \geq 3$. Consider the number field $K = \Bbb{Q}(\omega)$ and the ring of integers $\mathcal{O}_K = \Bbb{Z}[w]$ that has the usual integral basis
$$B = ...
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votes
3answers
187 views
The kernel of the reciprocity map in global class field theory
I am learning Class Field Theory by reading Milne's notes and Neukirch's book. There is a proof I can't find.
Let $K$ be a number field. One constructs the map $I_K \rightarrow G_K^\text{ab}$ using ...
5
votes
1answer
104 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 ...
5
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1answer
98 views
Discrete valuation ring extension such that $A[\pi]$ is not integrally closed
Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero.
Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring?
If not, ...
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0answers
95 views
Splitting of a polynomial modulo primes of a ring of integers
I'm trying to finish an exercise from Daniel Marcus's "Number Fields" book: #30(e) in chapter 3, page 91.
Here's the problem: say $\mathcal{O}_K$ is the ring of integers of a number field $K$ and let ...
5
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0answers
71 views
Connection between ramification in number fields and Clifford theory
Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
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0answers
110 views
Question about a proof on p70 in Cassels' Local Fields
I'm trying to read the proof of
COROLLARY. The only solutions of
$x^2+7=2^m$ ($x,m \in \mathbb{Z}$) (6.15)
have $m=3,4,5,7,15$.
I don't see why there could be a + in $y\pm \alpha$ ...
5
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0answers
189 views
Picard group of an order in an imaginary quadratic field as a quotient of the idele class group
If $K$ is an imaginary quadratic field, then one can show that the orders in $K$ are precisely the rings $\mathbb{Z}+f\mathscr{O}_K$ for $f\geq 1$, where $\mathscr{O}_K$ is the ring of integers of ...
5
votes
0answers
203 views
Compute the Unit and Class Number of a pure cubic field $\mathbb{Q}(\sqrt[3]{6})$
Find a unit in $\mathbb{Q}(\sqrt[3]{6})$ and show that this field has class number $h=1$.
I am done with the first part which is relatively simple:
Suppose that $\varepsilon$ is a unit in ...
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0answers
97 views
Ramification of an integral closure of $\mathbb{C}\{z\}$
Let $\mathbb{C}\{z\}$ be the ring of convergent series in one
variable over $\mathbb{C}$, $K$ the fraction field of
$\mathbb{C}\{z\}$, $E$ a Galois extension of $K$ and
$\mathcal{O}_{E}$ the integral ...
4
votes
3answers
215 views
Does any integral domain contain an irreducible element?
Let $R$ be an integral domain which is not a field.
Does $R$ necessarily have an irreducible element?
I suspect the answer is no, but I couldn't find an example showing that...
4
votes
2answers
213 views
Geometrical approaches to Class group of Dedekind Rings
I premise that I do not know anything about fractionary ideals and class groups of Dedekind domains. But I know the theory of divisors on regular schemes, as treated in Hartshorne. What I would like ...
4
votes
3answers
485 views
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as ...
4
votes
4answers
253 views
Generalizing Cauchy-Riemann Equations to Arbitrary Algebraic Fields
Can it be done?
For an arbitrary quadratic field $Q[\sqrt{d}]$, it's easy to show the equations are simply $ f_x = -\sqrt{d} f_y $, where $ f : Q[\sqrt{d}] \to Q[\sqrt{d}]$. I'm working on the case ...
4
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2answers
213 views
How often is an irreducible polynomial irreducible?
The question doesn't of course make sense as written in the title. Here is what I really mean:
Given a global field $k$ and an irreducible polynomial $P \in k[x]$
Is it true that $P$ is reducible ...
4
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1answer
487 views
Application of ideal class group?
I know what the class group is but I could not come up with any setting where we need to know what the class group actually is.
Does anyone know of an example where we have two number fields with the ...
4
votes
3answers
133 views
Show that there exists $f ∈ \mathbb{Z}$ such that $f^2 + f +1 ≡ 0 \pmod p$.
Let $p ≡ 1 \pmod 3$ be a prime. Show that there exists $f \in \mathbb{Z}$ such that $f^2 + f +1 \equiv 0 \pmod p$.
I know the first few primes of this form are: $7,13,19$
So for example $p=7$ we ...
4
votes
1answer
309 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 ...
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votes
3answers
489 views
Discriminant of a monic irreducible integer polynomial vs. discriminant of its splitting field
Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that ...
4
votes
4answers
128 views
Algebraic Number Theory - Lemma for Fermat's Equation with $n=3$
I have to prove the following, in my notes it is lemma before Fermat's Equation, case $n=3$. I was able to prove everything up to the last two points:
Let $\zeta=e^{(\frac{2\pi i}{3})}$. Consider ...
4
votes
2answers
137 views
Does this equation have integer solutions
Let $g\geq 2$ be an integer. (It will be the genus of some curve.)
Are there positive integers $d$ and $e$ such that the equality
$$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
4
votes
2answers
159 views
Show that a polynomial is irreducible in $\mathbb{Q}$
Show that polynomial $f\left( x \right) = x\left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 4} \right) - a$ is irreducible in $\mathbb{Q}$, where $a \in - 2 + ...
4
votes
3answers
627 views
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 ...
4
votes
2answers
145 views
Extensions of number fields which are unramified at all finite primes
Let $k$ be a totally real number field of degree $n$.
I'd like to know how I can determine whether or not there exists a quadratic field extension $L$ of $k$ such that the extension $L/k$ is ...
4
votes
2answers
134 views
how to translate local reciprocity with galois groups into local reciprocity with weil groups?
I am trying to understand certain aspects of the Weil group $W_K$ for a $p$-adic field K, in particular how it does interplay with local class field theory.
Let $L/K$ be a finite unramified ...
4
votes
3answers
175 views
norm of an algebraic number with abs value smaller than 1
Let $\alpha \in \mathbb{C}$ be an algebraic number and $\alpha = \alpha_1, \alpha_2,...,\alpha_n$ its conjugates and $N(\alpha) = \prod_i \alpha_i$ its norm.
Is it true that $|\alpha| < 1 ...
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3answers
95 views
Prove or disprove that $\phi(a^n - 1)$ is divisible by n
I have a proof for the case of $a$ being prime I believe, I think this is also true for $a$ composite since I ran a test for the first $100$ numbers over the first $100$ values of $n$ and it seems to ...
4
votes
3answers
112 views
Finding minimal polynomial of ${e^{{2\pi}i/3}} + 2$ over $\mathbb{Q}$
I'm trying to find the minimal polynomial of this. I tried setting ${e^{{2\pi}i/3}} + 2$ = $\alpha$ but got stuck there. Help would be greatly appreciated.
4
votes
2answers
403 views
Roots of unity in a local field
The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as
$K=\langle \pi\rangle\times \mu_{q-1}\times ...
4
votes
3answers
74 views
Proof of Hasse-Minkowski over Number Field
Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre's "A Course in Arithmetic" has a self contained proof ...
