Body compound

In mathematics , the compound of two bodies is their smallest joint upper body.

For terms used in this article (such as “body adjunct”, “intermediate body”, and “degree of expansion”), see body expansion .

Are and are the lower body of the body , then the body compound is defined as ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle K}$ ${\displaystyle AB}$

${\displaystyle AB:=A(B).}$

A ( B ) denotes the body adjunction of the set to the body , it consists of all fractions of - linear combinations of elements . The adjunction in this case is symmetrical, ie . ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A(B)=B(A)}$

If and are intermediate fields of a field extension , and if both are finite extensions of , then the degree of extension of the compound is at most equal to the product of the two individual degrees of extension and at least as large as their smallest common multiple (LCM): ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle L/K}$ ${\displaystyle K}$

${\displaystyle \operatorname {kgV} (\left[A:K\right],\left[B:K\right])\leq \left[AB:K\right]\leq \left[A:K\right]\cdot \left[B:K\right].}$

If and are linearly disjoint , then this is e.g. B. the case when the degrees of expansion of and are relatively prime . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \left[AB:K\right]=\left[A:K\right]\cdot \left[B:K\right].}$ ${\displaystyle A}$ ${\displaystyle B}$

You can also consider the compound of any number of parts of a common upper body. B. the field of algebraic numbers is an upper body of every finite extension of , and is equal to the compound of all finite extensions. ${\displaystyle \mathbb {Q} }$