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        Questions tagged [algebraic-number-theory]

        Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces

        3
        votes
        0answers
        98 views

        If a and b are roots of polynomials P and Q, then what polynomials are a+b and ab a roots of? [on hold]

        If $a \in \mathbb{R}$ and $b \in \mathbb{R}$ are roots of polynomials $P$ and $Q$ with rational coefficients, is there an algorithm / formula / process for finding polynomials with rational ...
        4
        votes
        5answers
        1k views

        Connection Between Knot Theory and Number Theory

        Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
        6
        votes
        0answers
        104 views

        Extension of Erdos-Selfridge Theorem

        Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem $\text{Theorem 1:}$ The product of two or more consecutive positive integers is ...
        10
        votes
        1answer
        772 views

        Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

        I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following: Lemma: if L/K is an abelian ...
        4
        votes
        0answers
        78 views

        Does the Gauss sum attached to $\chi$ ever belong to $\mathbb{Q}(\chi)$?

        Let $p$ be a prime number and $\chi$ be a primitive Dirichlet character of conductor $p$. We let $$g(\chi)=\sum_{a=1}^{p}{\chi(a)e^{2i\pi a/p}}$$ be the Gauss sum attached to $\chi$. Is this known ...
        1
        vote
        1answer
        131 views

        Bound on number of proper ideals of norm equal to n

        I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand: Let $d$ be a positive non-square interger and set let $K = \...
        4
        votes
        0answers
        87 views

        Hodge-Tate weights of cohomological cuspidal automorphic representation

        Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard ...
        4
        votes
        1answer
        230 views

        Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd?

        Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not ...
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        votes
        0answers
        66 views

        Absolute convergence of the Fourier series of a smooth adelic function

        Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...
        37
        votes
        1answer
        1k views

        Class field theory - a “dead end”?

        I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes ...
        4
        votes
        0answers
        73 views

        Computation of Hochschild homology

        Let $A$ be a Dedekind domain. Let $n\geq 2$ be an integer. Is there a simple description of $HH_*(A, A/nA)$?
        4
        votes
        1answer
        245 views

        Confusion about topological Hochschild homology and $\mathbb{Z}_p$-topological Hochschild homology

        Let $R$ be the ring of integers in a perfectoid field of mixed characteristic $p$. Is $\pi_*THH(R)$ (as defined in Bhatt--Morrow--Scholze) $p$-complete (as an abelian group)?
        6
        votes
        1answer
        397 views

        The Hilbert symbols of quaternion algebras over a totally real field

        Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as $$B = \left(\frac{a,b}{k}\right), $$ for some constants $a,b \in k^\times$. My question is, can I always ...
        9
        votes
        0answers
        170 views

        How small may the discriminant of an $S_d$-field be?

        In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
        5
        votes
        0answers
        160 views

        How did Gauss find the units of the cubic field $Q[n^{1/3}]$?

        Recently I read jstor article "Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas regarding the foundations of algebraic number ...

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