Calculator
Formula used
- Rational \(r = p/q\): minimal polynomial over \(\mathbb{Q}\) is \(x-r\). As an integer polynomial: qx - p = 0.
- Quadratic surd \(\alpha = a + b\sqrt{d}\): eliminate \(\sqrt{d}\) by squaring: (x-a)^2 - b^2 d = 0.
- Complex \(z = a + bi\): use conjugates so coefficients stay rational: (x-(a+bi))(x-(a-bi)) = (x-a)^2 + b^2.
- Given polynomial: if the polynomial is irreducible over \(\mathbb{Q}\), then it is the minimal polynomial of each of its roots.
How to use this calculator
- Select an Input type that matches your number.
- Enter exact values using p/q whenever possible.
- Click Compute Minimal Polynomial to see the result above the form.
- Review Computation details for the applied formula and simplifications.
- Use Download CSV or Download PDF for reports.
Example data table
| Input type | Example input | Output polynomial | Degree |
|---|---|---|---|
| Rational | 3/4 | 4x - 3 = 0 | 1 |
| Quadratic surd | a=1/2, b=1, d=2 | 4x^2 - 4x - 7 = 0 | 2 |
| Complex | a=1/3, b=2 | 9x^2 - 6x + 37 = 0 | 2 |
| Given polynomial | 1 0 -2 (x² − 2) | x^2 - 2 = 0 | 2 |
Why minimal polynomials matter
The minimal polynomial of an algebraic number is the unique monic polynomial in \(\mathbb{Q}[x]\) of smallest degree that has the number as a root. Its degree equals the dimension of \(\mathbb{Q}(\alpha)\) over \(\mathbb{Q}\), so it measures how many basis elements are needed to express powers of \(\alpha\). Knowing the polynomial lets you reduce higher powers using the relation \(m(\alpha)=0\), keeping symbolic work exact. It also provides the right recurrence for sequences like \(1,\alpha,\alpha^2,\ldots\), which is useful when building multiplication tables in a chosen basis.
Input coverage and degree outcomes
This calculator targets common classroom inputs: rationals \(p/q\), quadratic surds \(a+b\sqrt{d}\), complex rationals \(a+bi\), and a user-supplied polynomial. Typical outcomes are degree 1 (rational numbers) and degree 2 (quadratic extensions). For surds, choosing squarefree \(d\) avoids hidden simplifications; for complex inputs, \(b\neq 0\) ensures the conjugate is distinct, forcing degree two. For a supplied polynomial, the tool reports it in primitive integer form; minimality for a specific root holds when the polynomial is irreducible over \(\mathbb{Q}\).
Elimination steps used in computation
For \(p/q\), clearing denominators converts \(x-\tfrac{p}{q}=0\) to \(qx-p=0\). For \(a+b\sqrt{d}\), subtract \(a\) and square to remove \(\sqrt{d}\), giving \((x-a)^2-b^2d=0\); expanding yields explicit coefficients. If \(a\) or \(b\) are rational fractions, the tool clears denominators first so the final polynomial has integer coefficients. For \(a+bi\), multiplying conjugates preserves rational coefficients: \((x-(a+bi))(x-(a-bi))=(x-a)^2+b^2\).
Interpreting the polynomial you receive
The display is a primitive integer polynomial, meaning the greatest common divisor of its coefficients is 1. This removes arbitrary scaling and is convenient for reporting and downloads. It differs from the monic minimal polynomial only by a nonzero rational factor, so it encodes the same field and roots. A leading coefficient greater than 1 usually indicates a denominator was cleared (for example, \(3/4\) becomes \(4x-3\)).
Quality checks and practical uses
Validate by substitution: the evaluated value should simplify to exactly zero, not just “close.” The details panel helps confirm simplification choices such as factoring out a common divisor. Minimal polynomials support exact equality testing, simplify nested radicals, and provide a clean algebraic relation for documentation. CSV/PDF exports bundle the input, method, and final polynomial for sharing. This is handy when you need consistent formatting across multiple examples, or when you want to archive results with timestamps for later verification.
FAQs
1) Is the output always the monic minimal polynomial?
The tool outputs a primitive integer polynomial with the same roots as the monic minimal polynomial. Divide by the leading coefficient to get a monic form in \(\mathbb{Q}[x]\).
2) Why does a rational \(p/q\) return \(qx-p\)?
Over \(\mathbb{Q}\), the minimal polynomial is \(x-\tfrac{p}{q}\). Multiplying by \(q\) clears denominators, producing an equivalent integer polynomial \(qx-p\) with the same root.
3) What should I choose for \(d\) in \(a+b\sqrt{d}\)?
Prefer a squarefree positive integer \(d\). If \(d\) has a square factor, simplify the radical first; the minimal polynomial should be computed for the simplified expression.
4) Why is the complex case quadratic even when \(a\) is rational?
If \(b\neq 0\), the number is non-real and its conjugate is distinct. The minimal polynomial over \(\mathbb{Q}\) must include both roots, giving \((x-a)^2+b^2\) of degree 2.
5) Can I use the “Given polynomial” mode to prove minimality?
Only when the polynomial is irreducible over \(\mathbb{Q}\). If it factors, a root may satisfy a lower-degree factor, so you must factor or test irreducibility first.
6) How do I quickly validate the result?
Substitute the input back into the polynomial. For rationals and simple surds/complex numbers, the evaluation should simplify to exactly zero. The computation details summarize the elimination used.