Algebraic element

The terms algebraic and transcendent element appear in abstract algebra and generalize the concept of algebraic and transcendent numbers .

definition

Be an extension of the body , an element. Then is called algebraic about if there is a polynomial different from the zero polynomial with coefficients in that has as a zero. ${\displaystyle L/K}$ ${\displaystyle a\in L}$ ${\displaystyle a}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle a}$

An element from that is not algebraically over is called transcendent over . ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle K}$

Examples

• A complex number is an algebraic number if and only if it is an algebraic element in the field extension . ${\displaystyle \mathbb {C} /\mathbb {Q} }$
• The square root of 2 is algebraic over because it is a zero of the polynomial whose coefficients are rational. ${\displaystyle \mathbb {Q} }$ ${\displaystyle X^{2}-2}$
• The circle number and Euler's number are transcendent over . But they are algebraically over because they are defined as real numbers. More generally applies: ${\displaystyle \pi }$ ${\displaystyle e}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$
• Every element of the field is algebraically about because it is the root of the linear polynomial . ${\displaystyle a}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle X-a}$
• Every complex number that can be formed by rational numbers, the basic arithmetic operations ( addition , subtraction , multiplication and division ) as well as by taking the root (with natural root exponents) is algebraic about . ${\displaystyle \mathbb {Q} }$
• However, it follows from Galois theory that, conversely, there are algebraic numbers that cannot be represented in this way; compare Abel-Ruffini's theorem . ${\displaystyle \mathbb {Q} }$
• Over the field of p-adic numbers is algebraically (as limit the number of reciprocal faculties), because for is and is in included. ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle e}$ ${\displaystyle p>2}$ ${\displaystyle e^{p}}$ ${\displaystyle p=2}$ ${\displaystyle e^{4}}$ ${\displaystyle \mathbb {Q} _{p}}$
• If one forms the body of the formal Laurentreihen for any body , the formal variable is a transcendent element of this extension. ${\displaystyle K}$ ${\displaystyle K((X))}$ ${\displaystyle X}$

properties

The following conditions are equivalent for an element made of (an upper body of ): ${\displaystyle a}$ ${\displaystyle L}$ ${\displaystyle K}$

• ${\displaystyle a}$is algebraically about . ${\displaystyle K}$
• The body expansion is of finite degree , i. i.e., is finite-dimensional as a vector space. ${\displaystyle K(a)/K}$ ${\displaystyle K(a)}$ ${\displaystyle K}$
• ${\displaystyle K(a)=K[a]}$

It is the Ringadjunktion of at which all the elements of is that as a polynomial over can write. is its quotient field in and consists of all elements of which can be written as with polynomials and over ( not equal to the zero polynomial). ${\displaystyle K[a]}$ ${\displaystyle a}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle g(a)}$ ${\displaystyle g}$ ${\displaystyle K}$ ${\displaystyle K(a)}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle g(a)/h(a)}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle K}$ ${\displaystyle h(a)}$

This characterization can be used to show that the sum, difference, product and quotient of over algebraic elements are again algebraically over . The set of all over algebraic elements of forms an intermediate field of the extension , the so-called algebraic closure in . This term should not be confused with the algebraic closure of . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle L/K}$ ${\displaystyle L}$ ${\displaystyle K}$

Minimal polynomial

→ Main article : minimal polynomial

If algebraic is over , then there is exactly one normalized polynomial from with the smallest degree and zero , this is called “the minimum polynomial from over ”. It is also called an algebraic element of degree relative to . has a possible basis as a vector space over the dimension . So the degree of expansion of is also the same . ${\displaystyle a}$ ${\displaystyle K}$ ${\displaystyle K[X]}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle K}$ ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle K(a)}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle \{1,a,a^{2},\dotsc ,a^{n-1}\}}$ ${\displaystyle K(a)/K}$ ${\displaystyle n}$

example

${\displaystyle a={\sqrt {2}}+{\sqrt {3}}}$is an algebraic element of degree 4 above , for from ${\displaystyle \mathbb {Q} }$

${\displaystyle a^{2}=5+2{\sqrt {6}}}$

${\displaystyle a^{3}=11{\sqrt {2}}+9{\sqrt {3}}}$

${\displaystyle a^{4}=49+20{\sqrt {6}}}$

the minimal polynomial results

${\displaystyle X^{4}-10X^{2}+1}$,

thus a 4th degree polynomial. So there is a basis of as a vector space over . Another possible basis is , i. H., ${\displaystyle \{1,a,a^{2},a^{3}\}}$ ${\displaystyle \mathbb {Q} \left({\sqrt {2}}+{\sqrt {3}}\right)}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \{1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}\}}$

${\displaystyle \mathbb {Q} \left({\sqrt {2}}+{\sqrt {3}}\right)=\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \}}$

and is a degree 4 body extension. ${\displaystyle \mathbb {Q} \left({\sqrt {2}}+{\sqrt {3}}\right)/\mathbb {Q} }$

generalization

The concept of the whole element can be defined in ring extensions . If one understands a body expansion as a ring expansion, then an element is there exactly if it is an algebraic element of the body expansion.

Individual evidence

1. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , definition 6.2.10.
2. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , sentence 6.3.3 and sentence 6.3.4.
3. ↑ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , chapter 6.3.