# Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

4,794 questions
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### Dedekind theorem of ramification in cyclotomic fields

Let $\Bbb Q(w)$ denote the $n$-th cyclotomic field then$\Bbb Z[w]$ is its ring of integers, $d$ denotes the discriminant of $\Bbb Q(w)$, and $p\mid d$ then $p\Bbb Z$ must ramify in $\Bbb Q(w)$. In ...
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### Langlands L functions for groups over finite fields.

In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
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### A polynomial whose range gaps are the primes [closed]

Not sure what to add here. I responded to hardmath in TN e Peter. If you feel it doesn't generate discussion I will rethink the question and read the purpose of your platform and try to reform the ...
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### extension of discrete valued fields and ring of integers

Let $K$ be a complete discrete valued field with ring of integers $A$, maximal ideal $m$ and uniformizer $\pi_K$. Let $L$ be a finite extension of $K$ with ring of integers $B$, maximal ideal $m'$ ...
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### Does any fractional ideal of $R$ always contain a non-zero element of $R$?

Let $R$ be an integral domain. Let $A$ be a non-zero fractional ideal of $R.$ Then can we say that $A$ always contains a non-zero element of $R$? Please help me in this regard. Thank you very much.
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### Understanding the Hurwitz-Kronecker Class Number Formula

The Hurwitz-Kronecker Class Number is given by the formula $H(d)=\sum_{Q\in Q_d/(\Gamma=PSL_2(\mathbb{Z}))}\frac{1}{w_Q}$ where $w_Q=card(stab(\alpha_Q))$ with $\alpha_Q$ being the unique zero ...
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### Showing that ideals are principal in ring of integers of cubic field

Let $K = \mathbb{Q}(\sqrt{6})$. Factorise $\langle p \rangle$ into prime ideals in $\mathcal{O}_K$ for $p = 2, 5, 13$, checking that the factors are principal. I used the Dedekind-Kummer Theorem ...
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### Abelian extensions and the Artin map

I'm trying to understand a proof of the following: Let $L$ and $M$ be Abelian extensions of $K$ (a number field). Then $L\subset M$ if and only if there is a modulus $\mathfrak{m}$, divisible ...
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### Trace of an algebraic integer is an integer?

Let $F$ be a number field and let $\alpha \in F$. If $\alpha \in \mathcal{O}_F$, then it is known that $N(\alpha) \in \mathbb{Z}$. I was wondering if something similar can be said about the trace? ...
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### history of analogy between number field and function fields

it is well known that an analogy exist between number fields and function fields and you can translate ideas and problems about one of them to another. there are many problems in number theory ...
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### Viewing ramification in $p$-adic fields in the context of algebraic geometry

Let $f: X \rightarrow S$ be a morphism of schemes which is locally of finite type. We say $f$ is unramified if for every $x \in X$, we have $(\Omega_{X/S})_x = 0$, where $\Omega_{X/S}$ is the sheaf ...
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### Splitting primes and quadratic reciprocity.

I'm reading through Marcus's wonderful book Number Fields, and I had two questions on his proof of quadratic reciprocity in chapter $4$. Marcus states first that if $p$ is an odd prime, and $q$ is ...
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### How come algebraic closure is not unique?

I was getting confused trying to understand why an algebraic closure of a field is not unique. If I consider any rational polynomial in one variable, then the roots may not be in $\mathbb{Q}$ but aren'...
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### Showing integrality of a certain ring [duplicate]

Let $A\subset B$ be integral domains, and let $b\in B$ be a unit. I am trying to show that $A[b]\cap A[b^{-1}]$ is integral over A. Let $x\in A[b]\cap A[b^{-1}]$. It suffices to show that $A[x]$ is a ...
### Confused about the splitting of 2 in $\mathbb{Q}(i).$
How do we split 2 in the cyclotomic ring $S = \mathbb{Z}[\sqrt{-1}]?$ Clearly the field $\mathbb{Q}(i)$ is a degree 2 normal extension. However, $(2)$ in $S$ is equal to $(2) = (1 - i)^2.$ Thus, the ...