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Results tagged with algebraic-number-theory
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user 49003
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
16
votes
Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$
Squares of elements of the $3$-adic integers $\mathbb{Z}_3$ are congruent to $0$ or $1$ modulo $3$, thus they are all at $3$-adic distance $1$ from $2$.
Squares of elements of $\mathbb{Q}_3 \setminus …
- 2,391
14
votes
Accepted
Does the Galois group of a Pisot polynomial contain the alternating group?
David Speyer has beaten me to it, but for what it's worth, a simple explicit example is
$$p(X)=X^4-2X^3-5X^2-4X-1$$
which is a Pisot polynomial, and has Galois group the dihedral group of order 8.
Ho …
- 2,391
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)}{(x_ …
- 2,391
12
votes
Accepted
In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie...
Zero is the only algebraic integer which has all its conjugates strictly inside the complex unit circle. (Look at the norm.)
For explicit examples with conjugates on either side of the unit circle, yo …
- 2,391
11
votes
Accepted
Is it decidable whether two real algebraic irrationals generate the same extension of the ra...
Yes, there are algorithms for factoring $f_\beta$ over the number field $\mathbb{Q}(\alpha)$; if a linear factor is found and both irreducible polynomials have the same degree, then $\mathbb{Q}(\alpha …
- 2,391
9
votes
Can a sum of roots of unity be an integer?
EDITED Despite the upvotes, the first version of this answer was mostly wrong, and I cannot delete it since it has been accepted. Let me see what can be salvaged.
Fix $\zeta_n$, identify $(\mathbb{Z …
5
votes
unit group of biquadratic fields
Ad 1: The index is easily seen to be finite, and the details will depend on how, exactly, the Galois group acts on the large unit group---and this in turn correlates to some extent with how it acts on …
- 2,391
5
votes
Accepted
Units of an extension of $\mathbb{Z}$
If and only if $\mathbb{Q}(\theta)$ is a real quadratic field.
In the imaginary quadratic case, and when $P$ has degree $1$, there are only finitely many units in the ring of integers of the field. W …
- 2,391
4
votes
The outer automorphism of the dihedral group $D_4$ and quartic polynomials
Let $\alpha=\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the conjugates of $\alpha$ in $M$, numbered so that one of the 4-cycles in the Galois group permutes them in the order $\alpha_1 \mapsto \alpha_2 …
- 2,391
4
votes
Accepted
Discriminant of a radical extension of a quadratic number field
Write $D_n$ for the absolute discriminant $\left|\mathrm{disc}(L_n|\mathbb{Q})\right|$ of the field of interest and $d_n$ for its degree. Then the root discriminant is bracketed in the interval $$5^{3 …
- 2,391
4
votes
Accepted
A question on the genus field of an algebraic number field
The setup ensures that the compositum $k(\sqrt{i})$ will be (isomorphic to) $k\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{i})$.
When $g$ is $2$-adically close to $x^4+1$ -- it turns out that mod $8$ (i.e., …
- 2,391
3
votes
Orders of the conjugates of an algebraic prime number in its residue field
It depends on how large the image $U$ of the global units $\mathcal{O}_K^\times$ in the group $$\prod_\sigma (\mathcal{O}_K/(\sigma(q)))^\times$$
happens to be. It's almost never the full group (becau …
- 2,391
3
votes
Accepted
Are totally positive units of $L$ closed (in $L$) with respect to the finite-idelic topology?
I believe this is difficult in general, and I don't think it has been studied much yet.
A partial answer, showing where not to look for simple counterexamples:
Unwinding the definitions, everything …
- 2,391
3
votes
Accepted
Criterion for Ramification of ray class field $K(\mathfrak{p})$ in $\mathfrak{p}$
When the global units surject, then any generator of a principal ideal coprime to ${\mathfrak{m}}$ can always be multiplied by a unit to replace it with a generator of the same principal ideal that is …
- 2,391
2
votes
Distinct projections along factors of splitting prime
Let $f\in\mathbb{Z}[X]$ be the characteristic polynomial of $\alpha$. If the reduction of $f$ mod $p$ has any repeated factors in $\mathbb{F}_p[X]$, then $p$ divides the discriminant of $f$. This is t …
- 2,391