All Questions
Tagged with ideals number-theory
137
questions
6
votes
1
answer
217
views
Norm is multiplicative?
Let $L/K$ be a finite extension of number fields. Let $I$ be an ideal in $L$, we define the norm $N(I)$ of $I$ to be the ideal in $K$ generated by elements of the type $N_{L|K}(a)$ where $a \in I.$ I ...
- 751
4
votes
2
answers
474
views
How important is getting nitty-gritty with ideals for algebraic number theory?
Coming off an undergraduate course on number fields based on Marcus's textbook Number Fields, I am interested in taking the logical next step towards (local) class field theory, as well as Iwasawa ...
- 155
1
vote
0
answers
34
views
Can we define valuation(s) in any integral ring?
Let $R$ be an integral domain with unity, that is not a field, and $\mathfrak{m}$ a maximal ideal in $R$, that is not the $(0)$ ideal.
I am following a course in which I learned that when the ...
- 11
1
vote
0
answers
24
views
Sum of the restrictions of Z ideals to an interval
I am currently studying a combinatorics question that makes appear the following type of sets:
$p\mathbb{Z}\cap [n]+q\mathbb{Z}\cap[n]$.
It is basically interescting ideals of $\mathbb{Z}$, but ...
- 11
2
votes
1
answer
97
views
Fractional $\mathbb Z[\sqrt{-3}]$-ideals
I want to show that each fractional ideal of $\mathbb Z[\sqrt{-3}]$ is of the form $\mathbb Z[\sqrt{-3}]a$ or $\mathbb Z[1/2+1/2\sqrt{-3}]a$ with $a\in\mathbb Q(\sqrt{-3})$.
Initial considerations:
...
- 4,062
0
votes
1
answer
44
views
Clarification regarding statement in class field theory
Let $K$ be an imaginary quadratic field, $\mathcal{O}_K$ be its ring of integers, $\mathcal{O}$ be an order, $I_K, P_K$ be the group of ideals and principal ideals in $\mathcal{O}_K$, $I_K(m)$ be the $...
- 600
2
votes
1
answer
149
views
Find a non-trivial element in the class group of $\mathbb{Q}( \sqrt{−5})$.
a. Find a non-trivial element in the class group of $\mathbb{Q}(\sqrt{−5})$.
b. Show that the class group of $\mathbb{Q}( \sqrt{−5})$ has order two.
For part a: I know that the class group is the ...
- 81
2
votes
2
answers
139
views
Let $I=(3, \sqrt{-14}-1)$ be an ideal in $\mathbb{Z}[\sqrt{-14}].$ Prove that $I, I^2, I^3$ are not principal but $I^4$ is.
For $I$ and $I^2$ I can directly calculate the product and then apply the norm trick to get a contradiction that if we assume they are principal, but I'm wondering that if there is any other good way ...
- 23
0
votes
0
answers
72
views
All prime ideals of size less than the Minkowski bound being principal implies that the ring of integers is a PID
So I am currently learning about ideal class groups and it is mentioned by the professor that to "show that the ring of integers $R$ is a PID, it suffices to show that all prime ideals of size ...
- 428
1
vote
1
answer
91
views
Proof of Kummer's Theorem in Janusz's Algebraic Number Fields
There is a theorem in Janusz's Algebraic Number Fields stated as follows:
Kummer's Theorem:
Let $R$ be a Dedekind ring with quotient field $K$ and $R'$ the integral closure of $R$ in a finite ...
- 27
0
votes
0
answers
46
views
Canonical form of module in a real quadratic field
Consider a real quadratic field K. Let M be a complete Z-module in K.
I would like to see a proof that it can be multiplied by a totally positive number $x$ so that
$$xM=\mathbb{Z} + w\mathbb{Z}$$
...
- 1,509
0
votes
1
answer
155
views
Understanding the narrow class group
The narrow class group is the group of fractional ideal modulo totally positive principal ideals. In the case of quadratic fields, this is to say positive norms principal ideals.
Consider $\sqrt{7}$, ...
- 1,509
0
votes
0
answers
31
views
Independence of fractional ideal for representation numbers in real quadratic number fields
Let $K$ be a real quadratic number field of prime discriminant $D$. We define for $\mathfrak a$ being a fractional ideal of $K$, $n \in \mathbb Z$ and $b \in \mathbb N$
$$R(\mathfrak a,n,b) := \# \{ x ...
- 6,184
2
votes
0
answers
78
views
Trace of norm ideal
Let $I$ be a fractional ideal of a real quadratic number field $K$ of discriminant $D$. I thought a little bit about traces of ideals and want to ask if the following is correct. I have a proof for ...
- 6,184
4
votes
3
answers
140
views
Why is $(3, 1+\sqrt{-26})^3=( 1+\sqrt{-26})$ in $\mathbb Z[\sqrt{-26}]$?
$a = 3, b = 1+\sqrt{-26}$ then $(a,b)^3=(a^3,b^3,a^2b,ab^2)$
each generator except $a^3$ has a $b$ factor and $\bar b b=27$, so $"\subseteq"$. Now the question is how to obtain $b$ using these ...
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