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Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

Algebraic number theory is a branch of mathematics that deals with algebraic numbers, or algebraic structures related to integers. The algebraic numbers are roots of polynomials \begin{equation*} c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0 \end{equation*} with integer coefficients.

7689 questions
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What's the significance of Tate's thesis?

I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the functional equation. However, while I was able to…
Akhil Mathew
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Sums of roots of unity

If the integral linear combination of some $n$th roots of unity has magnitude 1, does this necessarily imply that this linear combination is some root of unity as well? More precisely, Let $\zeta_1, \ldots \zeta_k$ be $n$th roots of unity. If …
Name12345
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Serge Lang never explains anything

On page 149 of Algebraic Number Theory by Serge Lang, I'm trying to understand why the inclusion $$k^{\ast}N_k^K(J_K) \cap J_c \subseteq \psi^{-1}(P_c \mathfrak N(c))$$ is true. I've been trying for like 3 hours. Can anyone please explain? EDIT:…
Bless you
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Ring of integers in p-adic field

How do we compute the ring of integers in a finite extension of $\mathbb{Q}_p$? Say, for example, in $\mathbb{Q}_p(i)$. Over $\mathbb{Q}$ we would guess $\mathbb{Z}[i]$, compute the discriminant of this $\mathbb{Z}$ module and look for squares…
Anna B
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what does it mean for a prime at infinity to ramify?

I understand what it means for a prime number to ramify in a ring of integers of a number field. However, an infinite prime is an archimedean valuation, what does it mean for an archimedean valuation to ramify in a number field?
user7212
31
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Unramification of a prime ideal in an order of a finite Galois extension of an algebraic number field

Is the following proposition true? If yes, how would you prove this? Proposition Let $K$ be an algebraic number field. Let $L/K$ be a finite Galois extension. Let $A$ and $B$ be the rings of algebraic integers in $K$ and $L$ respectively. Let $G$ be…
Makoto Kato
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What do ideles and adeles look like?

I see the ideals of an algebraic number field as lattices and prime ideals are the ones which you can't refine. How can we form a picture of ideles and adeles?
user58512
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On the ring of integers of biquadratic number fields

This is Daniel A. Marcus, Number Fields, Exercise 2.29 If anyone can help with this problem, I'd greatly appreciate it. Let $K$ be the biquadratic field $\mathbb Q[\sqrt{m}, \sqrt{n}] = \{a + b\sqrt{m} + c\sqrt{n} + d\sqrt{mn}: a,b,c,d \in \mathbb…
MathMan1214
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Eisenstein and Quadratic Reciprocity as a consequence of Artin Reciprocity, and Composition of Reciprocity Laws

Question 1: I've heard that Eisenstein and Quadratic Reciprocity can be derived from the Artin Reciprocity by applying it to certain field extensions. But I haven't seen on any reference an explicit description of this, and I am here asking for…
Ash GX
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Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ (2) $\ p$ is split in $K\ \ \ \ \ \ $ if $\mathcal…
Phil-ZXX
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Every ideal of an algebraic number field can be principal in a suitable finite extension field

Let $K$ be an algebraic number field. Let $I$ be a non-zero ideal of the ring of integers $\mathcal{O}_K$ in $K$. By class field theory, there exists a finite extension(the Hilbert class field) $L$ of $K$ such that $I\mathcal{O}_L$ is principal. Can…
Makoto Kato
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Ramification in a tower of extensions

I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have: An extension $L/K$ and a some subextensions $E_1,\ldots,E_k$ sitting in between (without necessarily having…
dstt
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Intuition for Krasner's Lemma

From Milne's Algebraic Number Theory, we have (he assumes that $K$ is complete with respect to a discrete nonarchimedian absolute value, but I don't know where the discrete part is being used) Let $\alpha,\beta\in K^{al}$, and assume that $\alpha$…
DCT
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extension of Euler's totient function to number fields

It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as monoid) as the disjoint union $\coprod_{d|n}…
Amateuresque
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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 one differs from the other by a multiple of a…
Zev Chonoles
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