Advanced Calculator
Formula Used
Let F be a number field. Let r1 be the number of real embeddings. Let r2 be the number of complex conjugate embedding pairs. The degree check is:
degree(F) = r1 + 2r2
The calculator applies these rank rules for the ring of integers OF:
- n = 0: rank K0(OF) = 1
- n = 1: rank K1(OF) = r1 + r2 - 1
- n > 1 and n even: rank Kn(OF) = 0
- n ≡ 1 mod 4: rank Kn(OF) = r1 + r2
- n ≡ 3 mod 4: rank Kn(OF) = r2
The displayed rank means the dimension of Kn(OF) tensor Q. It does not calculate torsion.
Comparison Table
| n | n mod 4 | Rank | Case | Rule |
|---|---|---|---|---|
| 0 | 0 | 1 | K0 base rank | rank K0(O_F) = 1 |
| 1 | 1 | 0 | Dirichlet unit rank | rank K1(O_F) = r1 + r2 - 1 |
| 2 | 2 | 0 | Even positive index | rank K_n(O_F) = 0 for even n > 0 |
| 3 | 3 | 0 | n congruent to 3 modulo 4 | rank K_n(O_F) = r2 |
| 4 | 0 | 0 | Even positive index | rank K_n(O_F) = 0 for even n > 0 |
| 5 | 1 | 1 | n congruent to 1 modulo 4 | rank K_n(O_F) = r1 + r2 |
| 6 | 2 | 0 | Even positive index | rank K_n(O_F) = 0 for even n > 0 |
| 7 | 3 | 0 | n congruent to 3 modulo 4 | rank K_n(O_F) = r2 |
| 8 | 0 | 0 | Even positive index | rank K_n(O_F) = 0 for even n > 0 |
| 9 | 1 | 1 | n congruent to 1 modulo 4 | rank K_n(O_F) = r1 + r2 |
Example Data Table
| Example field type | r1 | r2 | Degree | n | Rank |
|---|---|---|---|---|---|
| Rational field Q | 1 | 0 | 1 | 5 | 1 |
| Real quadratic field | 2 | 0 | 2 | 5 | 2 |
| Imaginary quadratic field | 0 | 1 | 2 | 3 | 1 |
| Mixed cubic field | 1 | 1 | 3 | 7 | 1 |
| Totally real quartic field | 4 | 0 | 4 | 9 | 4 |
How to Use This Calculator
- Select a preset or choose a custom signature.
- Enter r1, the count of real embeddings.
- Enter r2, the count of complex conjugate embedding pairs.
- Enter the K-group index n.
- Add the optional field degree for validation.
- Set a comparison start index and row count.
- Press Calculate Rank to show the result.
- Use CSV or PDF to export the calculation.
Article: Borel Rank Calculation in K-Theory
Why Borel Rank Matters
Borel rank calculations help translate deep algebraic data into a clear integer. In algebraic K-theory, a number field has a signature. The signature is written as r1 and r2. The value r1 counts real embeddings. The value r2 counts pairs of complex embeddings. These two inputs control the rational rank of many K-groups of the ring of integers.
What the Tool Calculates
This calculator focuses on the rank of K_n(O_F). The symbol O_F means the ring of integers in the field F. The result is not a torsion order. It is the free rank after tensoring with the rational numbers. That makes the result useful for regulator dimensions, arithmetic models, and physics inspired computations.
The Modular Pattern
The pattern becomes especially elegant for higher indices. After K0 and K1, the rank depends on n modulo four. Even positive indices have zero rational rank. Indices congruent to one modulo four use r1 plus r2. Indices congruent to three modulo four use r2 only. This periodic structure gives a fast way to compare many groups.
Degree Validation
The degree check is also important. A number field with signature r1, r2 has degree r1 plus twice r2. The tool compares this implied degree with an optional degree value. It then warns you when the signature and degree disagree. This helps catch typing mistakes before exporting data.
Reading the Output
The result panel gives the main rank, the modular class, the implied degree, and a short interpretation. The comparison table then lists nearby K-indices. This is useful when building sequences for theoretical physics notes, spectral models, or arithmetic geometry references.
Export and Limits
Use the CSV export when you need spreadsheet data. Use the PDF export for a quick report. Both outputs include the same main calculation and comparison rows. The example table gives sample signatures, so you can test the tool quickly.
Educational Scope
This calculator is educational. It simplifies a difficult theorem into a controlled workflow. It does not compute torsion subgroups. It also does not identify a specific field from a polynomial. It assumes that r1 and r2 are already known. With accurate inputs, it gives a dependable rational rank guide.
For advanced work, keep separate notes about regulators, zeta values, and torsion. Those subjects are related, but need additional invariants and methods outside this rank tool.
FAQs
1. What does Borel rank mean here?
It means the rational free rank of K_n(O_F). The calculator reports the dimension after tensoring the K-group with Q. It does not measure torsion.
2. What are r1 and r2?
r1 is the number of real embeddings. r2 is the number of conjugate complex embedding pairs. Together they define the signature of a number field.
3. Why is degree equal to r1 plus 2r2?
Each real embedding contributes one. Each complex pair contributes two embeddings over the rationals. Their sum gives the field degree.
4. Does the calculator compute torsion?
No. It only computes rational rank. Torsion parts of algebraic K-groups need deeper methods and are outside this tool.
5. Why do even positive indices return zero?
In this Borel rank pattern, positive even K-groups have zero rational rank for rings of integers. They may still contain torsion.
6. What happens when n is congruent to 1 modulo 4?
For higher odd indices with n congruent to 1 modulo 4, the rank is r1 plus r2. Both real and complex contributions appear.
7. What happens when n is congruent to 3 modulo 4?
For higher odd indices with n congruent to 3 modulo 4, the rank is r2. Only complex embedding pairs contribute.
8. Can this identify a number field?
No. It does not parse polynomials or compute embeddings. You must already know the signature values r1 and r2.