Galois Module Tool Calculator

Calculator

This tool models a cyclic Galois-group action σ on a free module by an n×n matrix A. It reports invariants by solving (A−I)v=0 and checks A^k=I.

Coefficient system

Choose finite field arithmetic or exact rational fractions.

Prime modulus p

Used only for prime modulus mode (F_p).

Group order k

We test whether A^k equals the identity.

Module rank n

Matrix must be n×n. Keep n modest for clarity.

Options

Show matrices (A, A^k, A−I)

Show RREF output

Turn on extra output for deeper inspection.

Generator matrix A (CSV or space separated)

Enter n rows. In rational mode, you may use fractions like 3/5.

Example data table

This sample represents a cyclic action of order 3 over F₅ on a rank-2 module.

Parameter	Value	Meaning
k	3	Order of the cyclic group C₃
p	5	Prime modulus for coefficients in F₅
n	2	Rank of the module (dimension over the field)
A	[[0,1],[4,4]]	Matrix for σ, satisfying A³ = I (mod 5)

Formula used

Fixed submodule (invariants): M^G = { v ∈ M : σ·v = v }. With matrix A for σ, this becomes (A − I)v = 0.
Rank–nullity (field mode): dim ker(A−I) = n − rank(A−I). The tool reports both rank and nullity.
Order relation: For a cyclic group C_k, a valid action satisfies A^k = I. The calculator checks this relation directly.
Orbit cycle length: For any vector v, its orbit under σ has length equal to the smallest t ≥ 1 with A^tv = v.

How to use this calculator

Select a coefficient system: prime modulus field or rational coefficients.
Enter the group order k and the module rank n.
Provide the n×n generator matrix A for σ.
Click “Compute module invariants” to view results above the form.
Use the download buttons to export CSV or PDF reports.

Module action in one matrix

A Galois module records how a symmetry σ acts on a free rank‑n module by an n×n matrix A. In finite field mode, entries are reduced modulo a prime p, so arithmetic stays inside F_p. In rational mode, fractions are kept exact, which is useful for classroom examples. The tool treats σ as a generator of a cyclic group C_k and interprets repeated action as powers A^t.

Order checks for cyclic groups

For a genuine C_k representation, the defining relation is A^k = I. The calculator computes A^k by fast exponentiation and compares it with the identity matrix. A failing check is not “wrong”; it signals that the chosen k does not match the matrix order, or that the intended group is different. The determinant det(A) is also reported because non‑invertible A cannot represent group actions reliably.

Fixed vectors and invariants

Invariant elements satisfy σ·v = v, so they live in the kernel of (A−I). The calculator forms A−I, performs row reduction, and outputs rank(A−I) and nullity(A−I). Over a field, nullity equals dim ker(A−I), giving the dimension of the fixed subspace M^G. In prime field mode the tool additionally shows |M^G| = p^nullity, turning dimension into an explicit count of fixed vectors.

Orbit structure and cycle lengths

Beyond invariants, the action partitions basis vectors into orbits. For each standard basis vector e_i, the tool iterates A e_i, A² e_i, … until it returns to e_i or reaches k steps. The first return time is the cycle length and divides the group order when A^k = I. Comparing cycle lengths helps detect decompositions, such as direct sums of smaller cyclic submodules.

Reporting, teaching, and validation

The CSV and PDF exports bundle the summary, matrices, invariant basis vectors, and orbit lengths into a reusable record. This is handy for assignments, quick verification during proofs, and documenting computational experiments. When adjusting p, n, or k, you can compare reports side‑by‑side to see how invariants change. Use the example loader to start from a known action, then modify entries to explore stability and edge cases. RREF output can be toggled to audit elimination steps and catch entry mistakes early.

FAQs

1) What does this tool model, in practical terms?

It models a cyclic symmetry σ acting on a free module via a matrix A, then derives invariants, order checks, and orbit behavior from linear algebra over your chosen coefficient system.

2) Why must the modulus p be prime in finite field mode?

A prime modulus makes F_p a field, so every nonzero element has an inverse. Row reduction and determinant tests depend on inverses, which may fail in composite modulus arithmetic.

3) What does nullity(A−I) tell me?

Nullity(A−I) equals the dimension of the fixed subspace M^G over a field. It counts independent invariant directions; in prime mode, the tool also reports the exact number of fixed vectors p^nullity.

4) What if the report says A^k = I fails?

Then k is not compatible with the chosen generator matrix. Either adjust k to match the matrix order, or replace A with one satisfying the intended relation for your cyclic group.

5) Can I use fractions or negative entries?

Yes. Rational mode accepts integers and fractions like 3/5 and keeps them exact. Prime field mode accepts integers (including negatives), then reduces them modulo p automatically.

6) How are orbit cycle lengths computed here?

For each standard basis vector e_i, the tool repeatedly applies A and detects the first return to e_i. That return time is reported as the cycle length, capped at k steps.