theorems on continuation
Theorem 2. If the degree (http://planetmath.org/ExtensionField) of the field extension is and is an arbitrary exponent (http://planetmath.org/ExponentValuation2) of , then has at most continuations to the extension field .
Theorem 3. Let be an exponent valuation of the field and the ring of the exponent . Let be a finite extension and the integral closure of in . If are all different continuations of to the field and their rings (http://planetmath.org/RingOfExponent), then
The proofs of those theorems are found in , which is available also in Russian (original), English and French.
Corollary. The ring (of theorem 3) is a UFD. The exponents of , which are determined by the pairwise coprime prime elements of , coincide with the continuations of . If are the pairwise coprime prime elements of such that for all ’s and if the prime element of the ring has the
with a unit of , then is the ramification index of the exponent with respect to ().
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
|Title||theorems on continuation|
|Date of creation||2013-03-22 17:59:51|
|Last modified on||2013-03-22 17:59:51|
|Last modified by||pahio (2872)|
|Synonym||theorems on continuations of exponents|