How do you find the minimal polynomial?

The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα.

How do you find the minimal polynomial in linear algebra?

3 Answers

1. Hence x(x2−4) divides the minimal polynomial,
2. Hence all these implies that the minimal polynomial is either x(x2−4) or x2(x2−4).
3. 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 an annihilating polynomial?

A polynomial p(x) such that p(T) = 0 is called an annihilating polynomial for T, The monic polynomial pT(x) of least degree such that pT(T) = 0, is called the minimal polynomial of T. Any polynomial q(x) such that q(T) = 0 is said to annihilate (or kill) T.

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.

Is the endomorphism φ of a field diagonalizable?

An endomorphism φ of a finite dimensional vector space over a field F is diagonalizable if and only if its minimal polynomial factors completely over F into distinct linear factors.

Which is the minimal polynomial of a field?

For the minimal polynomial of an algebraic element of a field, see Minimal polynomial (field theory). In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μA.

Is the minimal polynomial E1 and χt the same?

This is in fact also the minimal polynomial μT and the characteristic polynomial χT: indeed μT,e1 divides μT which divides χT, and since the first and last are of degree 3 and all are monic, they must all be the same.

How is the Frobenius endomorphism related to field theory?

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields.