Minimal polynomial

A minimal polynomial is generally understood to be a polynomial of minimal degree that just fulfills a property that is no longer fulfilled by factors of smaller degree. In particular, in various sub-areas of mathematics, the minimal polynomial indicates the minimal linear dependency between the powers of a matrix or a linear mapping or, more generally, an element of an algebra .

definition

Let a body and a unitary - algebra . Then the minimal polynomial of an element is the normalized polynomial of the lowest degree, which has as a zero. ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle x\in A}$ ${\displaystyle x}$

The minimal polynomial can also be used as a normalized generator of the kernel of the homomorphism

${\displaystyle K[T]\to A,\quad a_{0}+a_{1}T+\cdots +a_{d}T^{d}\mapsto a_{0}+a_{1}x+\cdots +a_{d}x^{d}}$,

of the insertion homomorphism of , where the ring of polynomials with coefficients is off. ${\displaystyle x}$ ${\displaystyle K[T]}$ ${\displaystyle K}$

In a finite-dimensional algebra, each element has a unique minimal polynomial, in an infinite- dimensional algebra this does not have to be the case. There the elements that have a minimal polynomial are called algebraic elements over the basic field; Elements to which this does not apply are called transcendent elements.

Linear Algebra

The minimal a square - matrix on a body is the normalized polynomial of smallest degree with coefficients in so that (the zero matrix is). ${\displaystyle p}$ ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle p\left(A\right)=0}$

The following statements for out are equivalent : ${\displaystyle \lambda }$ ${\displaystyle K}$

• ${\displaystyle \lambda }$is zero of , d. H. , ${\displaystyle p}$ ${\displaystyle p\left(\lambda \right)=0}$
• ${\displaystyle \lambda }$is the root of the characteristic polynomial of , ${\displaystyle A}$
• ${\displaystyle \lambda }$is an eigenvalue of . ${\displaystyle A}$

The multiplicity of a zero of determines the length of the longest principal vector chain to the eigenvalue , i.e. i.e., the multiplicity is e.g. B. 4, then there is a chain of four linearly independent main vectors (of levels 1 to 4) to the eigenvalue . If there are further main vector chains for the eigenvalue , which are linearly independent of this chain of length 4, then they are by no means longer. Thus the size of the largest Jordan block belonging to the Jordan normal form of is identical to the multiplicity of in the minimal polynomial . ${\displaystyle \lambda }$ ${\displaystyle p}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle p}$

The geometric multiplicity of the eigenvalue of , on the other hand, means the number of linearly independent eigenvectors for this eigenvalue. In other words: the geometric multiplicity of an eigenvalue of the square matrix is the dimension of the solution space of . ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle \left(A-\lambda \cdot E\right)x=0}$

Somewhat more generally, one can examine the core of the insertion homomorphism of from the definition for an endomorphism of a vector space (even without fixing it on a certain basis ) ; this then leads to a minimal polynomial even with infinite-dimensional vector spaces if this core is not the zero vector space . A simple example are the projection mappings , which are idempotent by definition, i.e. they fulfill the relation . So every projection has one of the polynomials , or as a minimal polynomial. ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle F}$ ${\displaystyle P}$ ${\displaystyle P^{2}-P=0}$ ${\displaystyle p\left(x\right)=x^{2}-x}$ ${\displaystyle p\left(x\right)=x}$ ${\displaystyle p\left(x\right)=x-1}$

Body theory

In body theory , the minimal polynomial is a term that occurs when the body is expanded.

Be a field extension, the polynomial ring to with the indeterminate and be algebraic, that is, it exists with . Then there exists a polynomial (called the minimal polynomial) with the properties ${\displaystyle L/K}$ ${\displaystyle K[X]}$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle a\in L}$ ${\displaystyle 0\neq p(X)\in K[X]}$ ${\displaystyle p(a)=0}$ ${\displaystyle m(X)\in K[X]}$

1. ${\displaystyle m(X)}$ is standardized
2. ${\displaystyle m(a)=0}$
3. ${\displaystyle m(X)}$has minimal degree, i.e. H. applies ${\displaystyle \forall g(X)\in K[X]\setminus \{0\}}$ ${\displaystyle \deg(g)<\deg(m)\;\implies \;g(a)\neq 0}$
4. ${\displaystyle m(X)}$is unambiguous ( determined by), d. H. for every further one that fulfills properties 1–3 already applies ${\displaystyle a}$ ${\displaystyle m^{\ast }(X)\in K[X]}$ ${\displaystyle m^{\ast }(X)=m(X)}$

If one considers the extension field as a vector space over and a certain element as endomorphism (through the illustration ), one arrives at the same minimal polynomial (in the sense of linear algebra) as in the field theory for an algebraic element . ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle \alpha \in L}$ ${\displaystyle L}$ ${\displaystyle F_{\alpha }\colon L\to L,x\mapsto \alpha \cdot x}$ ${\displaystyle \alpha }$

properties

• Minimal polynomials are irreducible over the basic field.
• Every polynomial with coefficients in the base field that has an algebraic element as a zero is a (polynomial) multiple of the minimal polynomial of . ${\displaystyle x}$ ${\displaystyle x}$
• The degree of the minimal polynomial of is equal to the degree of the simple expansion . ${\displaystyle x}$ ${\displaystyle K(x)/K}$

See also: Disintegration bodies , Cayley-Hamilton theorem

Examples

• Consider the body extension with the imaginary unit : The minimal polynomial of is , because it has as a zero, is normalized, and every polynomial of smaller degree would be linear and only have one zero in . ${\displaystyle \mathbb {Q} (i)/\mathbb {Q} }$ ${\displaystyle \textstyle i}$
  ${\displaystyle \textstyle i}$ ${\displaystyle \textstyle x^{2}+1}$ ${\displaystyle \textstyle i}$ ${\displaystyle \mathbb {Q} }$
• The polynomial is not a minimal polynomial of any element of any extension, since it can be represented as and for none of its roots is a polynomial of the lowest degree . ${\displaystyle \textstyle x^{3}+x}$ ${\displaystyle (x^{2}+1)\cdot x}$

Examples of minimal polynomials of an algebraic element

• Minimal polynomials over from , where is some complex square root: is already a zero of . But this polynomial is irreducible over if . If so , then this is the minimum polynomial ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\sqrt {a}}}$ ${\displaystyle {\sqrt {a}}}$
  ${\displaystyle {\sqrt {a}}}$ ${\displaystyle \textstyle X^{2}-a}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\sqrt {a}}\notin \mathbb {Q} }$
  ${\displaystyle {\sqrt {a}}\in \mathbb {Q} }$ ${\displaystyle X-{\sqrt {a}}}$

• Minimal polynomials over from : It holds . So is zero of . However, this polynomial is not irreducible because it has the factorization . Obviously there is no zero of . So must be the zero of . And this polynomial is irreducible (e.g. by reduction modulo 2) ${\displaystyle \mathbb {Q} }$ ${\displaystyle \xi _{3}=\exp((2\pi i)/3)}$ ${\displaystyle \xi _{3}^{3}=1}$ ${\displaystyle \xi _{3}}$ ${\displaystyle X^{3}-1}$ ${\displaystyle (X-1)(X^{2}+X+1)}$
  ${\displaystyle \xi _{3}}$ ${\displaystyle X-1}$ ${\displaystyle \xi _{3}}$ ${\displaystyle X^{2}+X+1}$

• Minimal polynomial over von : Here it is helpful to consider a normal body extension , with . This is e.g. B. for given, the decay field of the polynomial . In , the minimal polynomial breaks down into linear factors. The zeros are conjugates of , i.e. of the form for a from the Galois group of . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \alpha ={\sqrt[{4}]{2}}+{\sqrt {2}}}$ ${\displaystyle L/\mathbb {Q} }$ ${\displaystyle \alpha \in L}$ ${\displaystyle L=\mathbb {Q} ({\sqrt[{4}]{2}},i)}$ ${\displaystyle X^{4}-2}$ ${\displaystyle L}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \sigma (\alpha )}$ ${\displaystyle \sigma }$ ${\displaystyle L/\mathbb {Q} }$

It is sufficient to determine the possible values (i.e. the conjugates of ). The minimal polynomial over of is what can be factored over . Thus the conjugates of exact ${\displaystyle \sigma (\alpha )=\sigma ({\sqrt[{4}]{2}}+{\sqrt {2}})=\sigma ({\sqrt[{4}]{2}})+\sigma ({\sqrt[{4}]{2}})^{2}}$ ${\displaystyle \sigma ({\sqrt[{4}]{2}})}$ ${\displaystyle {\sqrt[{4}]{2}}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\sqrt[{4}]{2}}}$ ${\displaystyle X^{4}-2}$ ${\displaystyle L}$ ${\displaystyle X^{4}-2=(X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}$ ${\displaystyle \alpha }$

${\displaystyle \alpha _{0}=\alpha }$,

${\displaystyle \alpha _{1}=-{\sqrt[{4}]{2}}+(-{\sqrt[{4}]{2}})^{2}=-{\sqrt[{4}]{2}}+{\sqrt {2}}}$,

${\displaystyle \alpha _{2}=i{\sqrt[{4}]{2}}+(i{\sqrt[{4}]{2}})^{2}=i{\sqrt[{4}]{2}}-{\sqrt {2}}}$ and

${\displaystyle \alpha _{3}=-i{\sqrt[{4}]{2}}+(-i{\sqrt[{4}]{2}})^{2}=-i{\sqrt[{4}]{2}}-{\sqrt {2}}}$.

The minimal polynomial of is thus ${\displaystyle \alpha }$

${\displaystyle (X-\alpha _{0})(X-\alpha _{1})(X-\alpha _{2})(X-\alpha _{3})=(X-{\sqrt[{4}]{2}}-{\sqrt {2}})(X+{\sqrt[{4}]{2}}-{\sqrt {2}})(X-i{\sqrt[{4}]{2}}+{\sqrt {2}})(X-i{\sqrt[{4}]{2}}-{\sqrt {2}})}$

${\displaystyle =X^{4}-4X^{2}-8X+2}$

literature

• Uwe Storch , Hartmut Wiebe: Textbook of Mathematics. For mathematicians, computer scientists and physicists. Volume 2: Linear Algebra. BI-Wissenschafts-Verlag, Mannheim et al. 1990, ISBN 3-411-14101-8 .
• Thomas W. Hungerford: Algebra (= Graduate Texts in Mathematics. Vol. 73). 5th printing. Springer, New York NY et al. 1989, ISBN 0-387-90518-9 .