## How do you find the minimal polynomial of a number?

If F = Q, E = R, α = √2, then the minimal polynomial for α is a(x) = x2 − 2. The base field F is important as it determines the possibilities for the coefficients of a(x). For instance, if we take F = R, then the minimal polynomial for α = √2 is a(x) = x − √2.

**What is meant by minimal polynomial?**

In linear algebra the minimal polynomial of an algebraic object is the monic polynomial of least degree which that object satisfies. Examples include the minimal polynomial of a square matrix, an endomorphism of a vector space or an algebraic number. The general setting is an algebra A over a field F.

### How do you find the minimal polynomial of a 4×4 matrix?

If d = 1, then mipoly = x + b, therefore, M + b*Id = 0 but this is not true. If d = 2, then mipoly = xˆ2 + ax + b, therefore, Mˆ2 + aM + bId = 0 after doing some calculations, I came to the conclusion that the system has no non-zero solutions since the rank of the matrix is 3.

**How do you find the Cyclotomic polynomial?**

with largest prime divisor p = pk, by repeated polynomial division, as detailed in Algorithm 1.

- Algorithm 1: Calculating Φn(z) by repeated division.
- Input: n = p1p2 ··· pk, a product of k distinct primes.
- Output: Φn(z), the nth cyclotomic polynomial.
- for j = 1 to k do.

#### How do you write a minimal polynomial?

For example, if A is a multiple aIn of the identity matrix, then its minimal polynomial is X − a since the kernel of aIn − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a)n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the …

**How do you find the characteristic and minimal polynomial?**

The characteristic polynomial of A is the product of all the elementary divisors. Hence, the sum of the degrees of the minimal polynomials equals the size of A. The minimal polynomial of A is the least common multiple of all the elementary divisors.

## How do you find a minimal polynomial example?

3 Answers

- Hence x(x2−4) divides the minimal polynomial,
- Hence all these implies that the minimal polynomial is either x(x2−4) or x2(x2−4).
- Now by putting the matrix in the equation x(x2−4) if it comes 0 then x(x2−4) is the minimal polynomial else x2(x2−4) is the minimal polynomial.

**What is the degree of minimal polynomial?**

The minimal polynomial is always well-defined and we have deg µA(X) ≤ n2. If we now replace A in this equation by the undeterminate X, we obtain a monic polynomial p(X) satisfying p(A) = 0 and the degree d of p is minimal by construction, hence p(X) = µA(X) by definition.