What is the minimal polynomial Theorem?

What is the minimal polynomial Theorem?

In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique.

How do you work out minimal polynomials?

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.

What is the degree of minimal polynomial?

[edit] Minimal polynomial of an algebraic number The minimal poynomial of an algebraic number α is the rational polynomial of least degree which has α as a root. The degree of the minimal polynomial of α is equal to the degree of the field extension Q(α)/Q.

What is primitive polynomial in information theory?

A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides xn − 1 is n = pm − 1.

What is a minimal field?

A minimal field of non-zero characteristic is algebraically closed. Minimal fields. DEFINITION 1. Let K be a field of characteristic p. We call K perfect if all its elements have a p-th root in K.

Why is minimal polynomial unique?

By the definition of minimal polynomial, deg(f)≤deg(g), where deg denotes degree. By Division Theorem for Polynomial Forms over Field, there exist polynomials q,r∈K[x] such that: We have now shown that f divides all polynomials in K[x] which vanish at α. By the monic restriction, it then follows that f is unique.

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.

What is minimal polynomial T?

A polynomial p(t) is called a minimal polynomial of T if p(t) is a monic polynomial of least positive degree for which p(T) = O i.e. the zero operator.

What does it mean for a polynomial to be reducible?

: a polynomial expressible as the product of two or more polynomials of lower degree.

What is polynomial code in data communication?

A polynomial code is a linear code having a set of valid code words that comprises of polynomials divisible by a shorter fixed polynomial is known as generator polynomial. They are used for error detection and correction during the transmission of data as well as storage of data.

What is a Monic formula?

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.

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.

Which is the only member of Jα with no minimal polynomial?

If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E / F . Minimal polynomials are useful for constructing and analyzing field extensions.

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.

Is the minimal polynomial of α irreducible over F?

This is the minimal polynomial of α with respect to E / F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E / F .