continuation of exponent
defined in the base field , is an exponent (http://planetmath.org/ExponentValuation) of .
Proof. The exponent of attains in the set also non-zero values; otherwise would be included in , the ring of the exponent . Since any element of are integral over , it would then be also integral over , which is integrally closed in its quotient field (see theorem 1 in ring of exponent); the situation would mean that and thus the whole would be contained in . This is impossible, because an exponent of attains also negative values. So we infer that does not vanish in the whole . Furthermore, attains in both negative and positive values, since .
Let be such an element of on which attains as its value the least possible positive integer in the field and let be an arbitrary non-zero element of . If
then , and thus on grounds of the choice of . This means that is always divisible by , i.e. that the values of the function in are integers. Because and , the function attains in every integer value. Also the conditions
are in , whence is an exponent of the field .
Definition. Let be a finite field extension. If the exponent of is tied with the exponent of via the condition (1), one says that induces to and that is the continuation of to . The positive integer , uniquely determined by (1), is the ramification index of with respect to (or with respect to the subfield ).
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
|Title||continuation of exponent|
|Date of creation||2013-03-22 17:59:49|
|Last modified on||2013-03-22 17:59:49|
|Last modified by||pahio (2872)|
|Synonym||prolongation of exponent|
|Defines||continuation of the exponent|
|Defines||ramification index of the exponent|