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23 votes

Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Not without an upper bound on the absolute value of the discriminant $\Delta$, because any finite list of $a_n$ amounts to a congruence condition on $\Delta$ that is satisfied by infinitely many $\...
Noam D. Elkies's user avatar
22 votes
Accepted

Explicit family of number rings $\mathcal{O}_{K_n}$ requiring $n$ generators?

An earlier question on this type of topic was asked by Zev Chonoles in 2010 at Which number fields are monogenic? and related questions and I want to draw your attention to the comment there by BCnrd ...
KConrad's user avatar
  • 49.6k
17 votes

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

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the ...
Venkataramana's user avatar
17 votes
Accepted

Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$

If $\zeta_K(1/2) \neq 0$ then $\zeta'_K(1/2) = 0$ if and only if $$ \log |D_K| = (\log(8\pi) + \gamma) n + \frac\pi2 r_1, $$ where $D_K$ is the discriminant of $K$, and $\gamma = 0.5772156649\ldots$ ...
Noam D. Elkies's user avatar
15 votes
Accepted

Determining the Mordell-Weil group of a universal elliptic curve

Specialize $a,b$ to functions giving the universal elliptic curve over the modular curve $X_0(N)$. These are known to have rank zero over the function field of the modular curve with coefficients over ...
Felipe Voloch's user avatar
15 votes

Does every ring of integers sit inside a monogenic ring of integers?

One reason this could be impossible is a local obstruction. Is it the case that, for some $p$, there are no degree $k$ extensions $L$ where $\mathcal O_L \otimes \mathbb Z_p$ is monogenic over $\...
Will Sawin's user avatar
  • 137k
15 votes
Accepted

How Dirichlet proved Dirichlet's unit theorem for general number fields?

Dirichlet's proof is described in Number Theory: Algebraic Numbers and Functions (starting on page 48). Dirichlet did not use Minkowski’s theorem; he proved the unit theorem in 1846 while ...
Carlo Beenakker's user avatar
14 votes

In which cyclic cubic number fields does there exist this type of unit?

The answer to question 1 is yes. Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio $$w=\frac{(x_1-1)(x_2-x_3)}{(...
GNiklasch's user avatar
  • 2,391
14 votes
Accepted

Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

More is true. Let $K/F$ be a field extension. Let $X$ be the vanishing set of some polynomials in $K[X_1,\dots, X_n]$. If $X$ contains a Zariski dense set of points with coordinates in $F$, then $X$ ...
Will Sawin's user avatar
  • 137k
12 votes
Accepted

Are the rationals definable in any number field?

According to R. S. Rumely, Undecidability and Definability for the theory of global fields, AMS Trans., 262, pp. 195-217, prime subfield is always definable in global field, and in number case, you ...
Denis T's user avatar
  • 4,446
12 votes

Sign and coefficients of fundamental unit of quadratic field

This might be useful: Stevenhagen, Peter, The number of real quadratic fields having units of negative norm, Exp. Math. 2, No. 2, 121-136 (1993). ZBL0792.11041. As Stevenhagen explains, if the ...
David Loeffler's user avatar
12 votes
Accepted

irreducibility of the polynomial $ x^4 +1 $

Among $-1$, $2$, and $-2$, there are $0$, $1$ or $3$ squares. This determines the irreducible factorization of $X^4+1$. If none is a square, it is irreducible; If only $-1=i^2$ is a square, the ...
YCor's user avatar
  • 60.4k
11 votes

Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?

Asking in terms of $B$ how many $a_n$ are needed is equivalent to asking the following question: What is the largest $N$ such that there exists two quadratic characters $\chi_1, \chi_2$ of ...
Will Sawin's user avatar
  • 137k
11 votes
Accepted

Upper bounds for regulators of real quadratic fields

Stephane Louboutin has several papers on getting explicit bounds for $L(1,\chi)$, for $\chi$ a character $\pmod q$. They're all of the strength of $1/2 \log q + $ an explicit constant. Some of his ...
Lucia's user avatar
  • 43.3k
11 votes

Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

I don't want to toot my own horn, but I coauthored a paper on this topic with Marvin Jones. One direction of our result is conditional on the Birch and Swinnerton-Dyer conjecture (see the first remark ...
Jeremy Rouse's user avatar
11 votes
Accepted

Irreducibility of polynomials over some number fields

Lemma. Let $K$ be any number field, and $p$ a prime unramified in $K$. Then $X^n-p$ is irreducible over $K$. Proof. It suffices to show that the field $L = K(\sqrt[n\ \ ]{p})$ has degree $n$ over $K$. ...
R. van Dobben de Bruyn's user avatar
10 votes
Accepted

class number of prime degree field with prime conductor

Maybe I am making a mistake here, but let me try: Let $H$ be the Hilbert class field of $K$. Then $H\cap \mathbb{Q}(\zeta_p)=K$ as otherwise one prime in there should be totally ramified and ...
Chris Wuthrich's user avatar
10 votes
Accepted

Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Barrucand and Cohn (MR0249396, Note on primes of type $x^2+32y^2$, class number, and residuacity. J. Reine Angew. Math. 238, 1969, 67--70) have proved that for primes $p \equiv 1 \textrm{ mod } 8$, ...
François Brunault's user avatar
10 votes
Accepted

What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

There is an existential definition of $\mathbb{Z}$ in the rational function field $\mathbb{R}(t)$ by a beautiful result of Denef using elliptic curves (Proposition 2 of The diophantine problem for ...
Arno Fehm's user avatar
  • 2,041
10 votes
Accepted

Is it true that this ideal must be principal? (proof verification)

I claim that under the given circumstances $\mathfrak{P}$ is not necessarily principal, i.e., the statement claimed in the question is wrong. Here is a counterexample. Consider $K = \mathbb{Q}$ and $L ...
AlexIvanov's user avatar
10 votes
Accepted

Algorithm for computing whether a cubic field is monogenic?

In the paper "Computing all power integral bases of cubic fields" (by Gaal and Schulte, published in Mathematics of Computation in 1989) the authors give an algorithm to determine if a cubic ...
Jeremy Rouse's user avatar
9 votes

Determining the Mordell-Weil group of a universal elliptic curve

If you want to do this directly, you could "partially specialize" to, say $y^2 = x^3 + Ax + T$ with $A\in\mathbb C$. Then I don't think it's very hard to show, via a standard descent, that as an ...
Joe Silverman's user avatar
9 votes
Accepted

Fields in which $ -1 $ can't be written as sum of two square elements

In the notation of Lam's Quadratic forms over fields, the Stufe (a German word) or level (its English translation) $s(F)$ of a field is the minimal $n$ such that $-1$ is the sum of $n$ squares. A ...
PseudoNeo's user avatar
  • 477
9 votes

A cyclic Galois extension over $ \mathbb{Q}(\omega)$

The answer is ``yes". Note first that in the question we may without loss restrict to cyclic extension of $2$-power degree, since the odd part of the order would split off as a direct factor, ...
Joachim König's user avatar
9 votes
Accepted

Questions about ray class groups

1. Surely, by "class group modulo $\mathfrak{f}$", the authors mean "ray class group modulo $\mathfrak{f}$". I guess the authors omit "ray", because $K$ has no real ...
GH from MO's user avatar
  • 99.4k
8 votes
Accepted

The Genus field and Hilbert class field

Let $K$ be a quadratic field with class group $\mathrm{Cl}(K)$ such that $\mathrm{Cl}(K)\neq \mathrm{Cl}(K)[2]$. Since the Galois group of the genus field is isomorphic to $\mathrm{Cl}(K)[2]$ and the ...
Aurel's user avatar
  • 4,933
8 votes
Accepted

Commutator subgroup of the absolute Galois group - a closed subgroup

No, the abstract commutator subgroup $[G_K,G_K]$ of the absolute Galois group $G_K$ of a number field $K$ is never closed: Write $[G,G]$ for the commutator subgroup of $G$ as an abstract group, and $c(...
Arno Fehm's user avatar
  • 2,041
8 votes

Galois embedding question for dihedral groups

The answer is "no", in general, since there may be local obstructions. Suppose, for example, that $k$ and $n$ are odd prime powers, and let $L/\mathbb{Q}$ be the unique intermediate ...
Alex B.'s user avatar
  • 12.8k
8 votes
Accepted

Standard conjecture on u-invariants?

For the classical $u$-invariant of fields of characteristic $\neq 2$, some known results are: The $u$-invariant of formally real fields is $\infty$. If $K$ is an algebraically closed field, its $u$-...
F Zaldivar's user avatar
  • 1,422
7 votes
Accepted

p-adic expansion for elements in algebraic closure of p-adic numbers

The following article discusses p-adic expansions in $\overline{\mathbb{Q}}_p$ and of $\mathbb{C}_p$. Algebraic $p$-adic expansions, David Lampert, Journal of Number Theory 23 (1986), 279–284.
Joe Silverman's user avatar

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