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, Homogeneous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
2,445 questions
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Is this proof that $A^3+B^3=C^n$ has no primitive solutions correct? [closed]
I am an independent researcher. This arose in the context of studying the Beal conjecture.
Setup: Factor $A^3+B^3=(A+B)(A^2-AB+B^2)=C^n$. For coprime $A,B$: $\gcd(A+B, A^2-AB+B^2)$ divides $3$. This ...
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Fallback for failure case in Galois factoring with units
Let $N = x^2 + 3y^2$ be a composite integer with a known representation of this form. Consider the cubic polynomial
$$
f(t) = 4t^3 - 3Nt - Nx,
$$
and let $K = \mathbb{Q}(\alpha)$ be the cubic number ...
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1
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426
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Properly transfinitely Euclidean rings of integers
For arbitrary integral domains, it is shown that the class of minimal order types of the images of the transfinite (i.e., $\mathrm{Ord}$-valued) Euclidean functions is the class of indecomposable ...
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Reduction map from infinite units of $K$ to finite ring
Let $K = \mathbb{Q}(\alpha) \cong \mathbb{Q}[x]/(f)$ be a cyclic number field of degree $n > 1$ over $\mathbb{Q}$, with ring of integers $\mathcal{O}_K$, and nonzero unit rank, with $\text{Gal}(K/\...
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Is Ramanujan's $G_{1255}$ connected to the snub dodecadodecahedron?
I. Polyhedra
In this MO question and the table below it, we checked all 75 uniform polyhedra and found that of the 12 uniform snub polyhedra, then TEN have Cartesian coordinates involving constants ...
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3
votes
2
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215
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Unit lattice in Galois closure, embeddings from subfields
Let $K$ be a number field. Let $L$ be the normal closure of $K$.
Since $L$ is Galois, it has many automorphisms. If the Galois group has many cosets, there are potentially many automorphisms that ...
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4
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3
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436
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Units in Galois quintic number field, discriminant/HT90
Let $K$ be a Galois quintic number field.
We can find many examples with small conductor (every abelian extension of $\mathbb{Q}$ is a subfield of some cyclotomic field) from LMFDB. All the examples ...
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5
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2
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368
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How to calculate local units mod global units?
For a finite Galois extension $K/\mathbb{Q}$ and a prime $p$, let $U= \Pi_{v|p}U_v$ be its local units, where $U_v$ is the unit group in the completion $K_v$. Let $E$ be $K$'s unit group, $\bar{E}$ ...
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4
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156
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Why $\ell=2,3$ is majorly studied for $\mathbb Z_\ell$-extension?
Disclaimer: This is migration of my Math.StackExchange post.
I am a master's student majoring in algebraic number theory. Last month, I (and my professor) found a topic that could be the subject of ...
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3
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1
answer
339
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Primes of the form $x^2+ny^2$ when the class group is $C_2C_4$
In my previous question, I asked about the quadratic forms of class number 8. While it answered my question about how to get the correct splitting polynomial, I'm still curious about conditions on the ...
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1
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0
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158
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Why is there only one Frobenius lift on $\mathbb{Z}_p$
I want to verify that there exists a unique $\delta$-structure on $\mathbb{Z}_p$.
Since $\mathbb{Z}_p$ has no torsion, we have a bijective correspondence between $\delta$-structures and lifts of ...
4
votes
1
answer
347
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Lawrence-Venkatesh for multi-term $S$-unit equations
I recently took another look at Lawrence and Venkatesh's well-known paper Diophantine equations and $p$-adic period mappings, and recalled that one of the elementary consequences is the finiteness of ...
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6
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1
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393
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Effective lower bound for class numbers of imaginary quadratic fields beyond Goldfeld-Odlyzko/Gross-Zagier
One of the most wondrous sagas in number theory is the path to solving Gauss's class number one oproblem for imaginary quadratic fields. For a given positive integer $d$, let $h(-d)$ be the class ...
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9
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1
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324
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Primes of the form $p=x^2+ny^2$ when the class number is 8
According to Cox (Primes of the form $x^2+ny^2$, Theorem $9.2$): Given a positive integer $n$, there exists a polynomial $f_n(x)$ of degree $h(-4n)$ which splits completely modulo $p$ if and only if $...
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9
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1
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Are two algebraic extensions with same polynomial root existence sentences isomorphic？
This is a question which may be well-known to experts but obscure to me.
Suppose we have a field $F$, let $\bar F$ be (one of) its algebraic closure. Given two algebraic extensions $K/F,\ L/F$ ...
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