Calculator
Example Data Table
| n = [K₁:ℚ] | m = [K₂:ℚ] | d = [K₁∩K₂:ℚ] | [K₁K₂:ℚ] = n·m/d | [K₁K₂:K₁] = m/d | [K₁K₂:K₂] = n/d |
|---|---|---|---|---|---|
| 3 | 5 | 1 | 15 | 5 | 3 |
| 4 | 6 | 2 | 12 | 3 | 2 |
| 8 | 12 | 4 | 24 | 3 | 2 |
Formula Used
For two finite extensions K₁/ℚ and K₂/ℚ, the compositum field is K₁K₂. If you know:
- n = [K₁:ℚ]
- m = [K₂:ℚ]
- d = [K₁∩K₂:ℚ]
Then the degree of the compositum is:
Related degrees follow from tower laws: [K₁K₂:K₁] = m/d and [K₁K₂:K₂] = n/d.
How to Use This Calculator
- Enter the degrees n and m.
- Enter the intersection degree d (at least 1).
- Press Submit to show results above the form.
- Use Download CSV or Download PDF if needed.
Practical meaning of a compositum degree
When two algebraic extensions are adjoined, the compositum captures the smallest field containing both. The degree [K₁K₂:ℚ] measures how many basis elements are needed to represent every element of the combined field. In computations, this degree acts like a scaling factor for complexity: larger degrees typically imply bigger multiplication tables, larger minimal polynomials, and heavier symbolic workloads.
How overlap changes the combined size
If the extensions share a nontrivial intersection, the overlap reduces the compositum degree. The calculator models this by an intersection degree d = [K₁∩K₂:ℚ]. Holding n and m fixed, doubling d halves [K₁K₂:ℚ]. This is a quantitative way to express “dependency” between the two extensions.
Key identities used by the calculator
The core identity is [K₁K₂:ℚ] = (n·m)/d. From tower laws, the relative degrees satisfy [K₁K₂:K₁] = m/d and [K₁K₂:K₂] = n/d. These ratios are useful for estimating how many new generators are needed when extending one field by elements from the other.
Choosing plausible inputs in practice
In many common settings, d divides both n and m, because intersection degrees often correspond to shared subextensions. If you enter a d that does not divide n or m, the tool still computes the algebraic ratio but shows a warning, since the triplet may not describe a realizable configuration.
Interpreting outputs for planning and learning
Use the compositum degree to compare scenarios: a disjoint case (d=1) gives the maximum combined degree n·m. Increasing d models shared structure, lowering the combined degree and the relative degrees over each field. This supports quick “what-if” analysis, and it reinforces how intersections control the effective size of adjoining extensions.
FAQs
1) What does the compositum field represent?
It is the smallest field that contains both extensions at once. Every element from either field can be expressed inside the compositum.
2) Why does the intersection degree reduce the result?
Shared substructure means fewer new basis elements are needed. The overlap is captured by d, which divides the product n×m in the degree formula.
3) What happens when d equals 1?
The tool treats the extensions as linearly disjoint over ℚ. The compositum degree becomes n×m, which is the maximum possible under this model.
4) Must d divide both n and m?
Often yes in standard algebraic constructions, but not always in abstract input experiments. The calculator warns if divisibility fails because the configuration may be unrealizable.
5) What do the relative degrees mean?
They measure how much larger the compositum is than each field. Specifically, [K₁K₂:K₁]=m/d and [K₁K₂:K₂]=n/d under the provided inputs.
6) How should I use this in study or work?
Use it for quick comparisons across cases and for validating intuition. Try changing d to see how overlap affects the combined degree and each relative extension size.