# characteristic polynomial of algebraic number

Let $\vartheta $ be an algebraic number^{} of degree $n$, $f(x)$ its minimal polynomial^{} and

$${\vartheta}_{1}=\vartheta ,{\vartheta}_{2},\mathrm{\dots},{\vartheta}_{n}$$ |

its algebraic conjugates.

Let $\alpha $ be an element of the number field^{} $\mathbb{Q}(\vartheta )$ and

$$r(x):={c}_{0}+{c}_{1}x+\mathrm{\dots}+{c}_{n-1}{x}^{n-1}$$ |

the canonical polynomial of $\alpha $ with respect to $\vartheta $. We consider the numbers

$r({\vartheta}_{1})=\alpha :={\alpha}^{(1)},r({\vartheta}_{2}):={\alpha}^{(2)},\mathrm{\dots},r({\vartheta}_{n}):={\alpha}^{(n)}$ | (1) |

and form the equation

$$g(x):=\prod _{i=1}^{n}[x-r({\vartheta}_{i})]=(x-{\alpha}^{(1)})(x-{\alpha}^{(2)})\mathrm{\cdots}(x-{\alpha}^{(n)})={x}^{n}+{g}_{1}{x}^{n-1}+\mathrm{\dots}+{g}_{n}=\mathrm{\hspace{0.33em}0},$$ |

the roots of which are the numbers (1) and only these. The coefficients^{} ${g}_{i}$ of the polynomial^{} $g(x)$ are symmetric polynomials^{} in the numbers ${\vartheta}_{1},{\vartheta}_{2},\mathrm{\dots},{\vartheta}_{n}$ and also symmetric polynomials in the numbers ${\alpha}^{(i)}$. The fundamental theorem of symmetric polynomials implies now that the symmetric polynomials ${g}_{i}$ in the roots ${\vartheta}_{i}$ of the equation $f(x)=0$ belong to the ring determined by the coefficients of the equation and of the canonical polynomial $r(x)$; thus the numbers ${g}_{i}$ are rational (whence the degree of $\alpha $ is at most equal to $n$).

It is not hard to show (see the entry degree of algebraic number^{}) of that the degree $k$ of $\alpha $ divides $n$ and that the numbers (1) consist of $\alpha $ and its algebraic conjugates ${\alpha}_{2},\mathrm{\dots},{\alpha}_{k}$, each of which appears in (1) exactly $\frac{n}{k}=m$ times. In fact, $g(x)={[a(x)]}^{m}$ where $a(x)$ is the minimal polynomial of $\alpha $ (consequently, the coefficients
${g}_{i}$ are integers if $\alpha $ is an algebraic integer^{}).

The polynomial $g(x)$ is the *characteristic polynomial ^{}* (in German

*Hauptpolynom*) of the element $\alpha $ of the algebraic number field $\mathbb{Q}(\vartheta )$ and the equation $g(x)=0$ the

*characteristic equation*(

*Hauptgleichung*) of $\alpha $. See the independence of characteristic polynomial on primitive element

^{}.

So, the roots of the characteristic equation of $\alpha $ are ${\alpha}^{(1)},{\alpha}^{(2)},\mathrm{\dots},{\alpha}^{(n)}$. They are called the $\mathbb{Q}(\vartheta )$*-conjugates ^{}* of $\alpha $; they all are algebraic conjugates of
$\alpha $.

Title | characteristic polynomial of algebraic number |
---|---|

Canonical name | CharacteristicPolynomialOfAlgebraicNumber |

Date of creation | 2013-03-22 19:08:41 |

Last modified on | 2013-03-22 19:08:41 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 14 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 15A18 |

Classification | msc 12F05 |

Classification | msc 11R04 |

Related topic | RationalIntegersInIdeals |

Related topic | DegreeOfAlgebraicNumber |

Defines | characteristic polynomial |

Defines | characteristic equation |

Defines | $\mathbb{Q}(\vartheta )$-conjugates |

Defines | $K$-conjugates |