Calculator
Example data table
| Mode | Inputs | Key output | Meaning |
|---|---|---|---|
| Cyclotomic | m=11, a=2 | sigma_2(zeta_11)=zeta_11^2 | Frobenius at primes p≡2 mod 11 acts as power-2. |
| Cyclotomic | m=8, a=3 | order(sigma_3)=2 | Automorphism repeats after two compositions. |
| Quadratic | d=5, n=3 | (D/n) = (5/3) = -1 | 3 is inert in Q(sqrt(5)); Frobenius is nontrivial. |
| Quadratic | d=-1, n=5 | (D/n) = (-4/5) = 1 | 5 splits in Q(i); Frobenius is identity. |
Formula used
1) Cyclotomic reciprocity over Q
For the abelian extension Q(zeta_m)/Q, the Artin map sends a residue class a (mod m) with gcd(a,m)=1 to the automorphism sigma_a defined by:
sigma_a(zeta_m) = zeta_m^a
The order of sigma_a is the multiplicative order of a in (Z/mZ)*.
2) Quadratic reciprocity as an Artin symbol
For K=Q(sqrt(d)) with discriminant D, the Frobenius at an unramified odd prime p is determined by the Kronecker symbol:
(D/p) in {+1, -1}
If (D/p)=+1, then p splits; if -1, then p is inert; and if the symbol is 0, then p ramifies.
How to use this calculator
- Select a mode: cyclotomic or quadratic.
- Enter integers that satisfy the stated conditions.
- Press Submit to compute the Artin image and invariants.
- Review the result card displayed above the form.
- Download CSV or PDF to keep a computation log.
FAQs
1) What does this tool compute in cyclotomic mode?
It computes the Artin image of a residue class a modulo m as an automorphism sigma_a with sigma_a(zeta_m)=zeta_m^a. It also reports phi(m) and the order of sigma_a in the Galois group.
2) Why must gcd(a, m) equal 1?
Only classes coprime to m define elements of (Z/mZ)*, which matches the Galois group of Q(zeta_m)/Q. If gcd(a,m)≠1, the corresponding primes are ramified and the Frobenius class is not defined.
3) What is the order of sigma_a?
It is the smallest positive integer k such that sigma_a^k is the identity. In cyclotomic fields this equals the multiplicative order of a modulo m, meaning a^k ≡ 1 (mod m).
4) What should I enter for d in quadratic mode?
Use a nonzero squarefree integer d, like -1, 2, -5, 5, 13. Squarefree inputs correspond to a quadratic field Q(sqrt(d)) without extra repeated prime factors.
5) Why does the tool compute a discriminant D?
In quadratic fields, reciprocity statements are cleanest with the field discriminant D. The splitting of primes is controlled by the Kronecker symbol (D/p), not directly by d in every case.
6) What does (D/n)=0 mean?
It indicates ramification: n shares a prime factor with the discriminant D. Ramified primes do not have a well-defined Frobenius element, so the tool reports that the Artin symbol is undefined there.
7) Can n be composite in quadratic mode?
Yes. The Kronecker symbol extends the Jacobi symbol to all integers. For composite n, the result gives a character value that is multiplicative across prime powers, useful for congruence and reciprocity checks.
8) Is this a full class field theory implementation?
No. It provides accurate computations for key abelian examples over Q: cyclotomic fields and quadratic fields. These are the standard entry points where Artin reciprocity can be computed explicitly and verified quickly.