determining the continuations of exponent
Task. Let be the 3-adic (triadic) (http://planetmath.org/PAdicValuation) exponent valuation of the field of the rational numbers and let be the ring of the exponent. Determine the integral closure of in the extension field and the continuations of to this field.
Any number of the quadratic field is of the form
satisfied by , belong to the ring , whence one has
The first of these inequalities implies that since is a unit of . As for , if one had , then , and therefore one had
Thus we have to have , too. So we have seen that for , it’s necessary that . The last condition is, apparently, also sufficient. Accordingly, we have obtained the result
Since the degree (http://planetmath.org/Degree) of the field extension is 2, the exponent has, by the theorem in the parent entry (http://planetmath.org/TheoremsOnContinuation), at most two continuations to . Moreover, the same entry (http://planetmath.org/TheoremsOnContinuation) implies that the intersection of the rings of those continuations coincides with , whose non-associated (http://planetmath.org/Associate) prime elements determine the continuations in question.
We will show that there are exactly two of those continuations and that one may choose e.g. the conjugate numbers
for such prime elements.
Suppose that splits in into factors (http://planetmath.org/DivisibilityInRings) as
where , (). Then also
where , . We perceive that
but according to the entry ring of exponent, the only prime numbers of are the associates of 3. Now we have factorised the prime number of into a product of two factors (http://planetmath.org/Product) and , and consequently, e.g. is a unit of and hence of , too. Thus and are units of , which means that and have only trivial factors. The numbers and themselves are not units, because ; and are not associates of each other, since . So and are non-associated prime elements of . This ring has no other prime elements non-associated with both and , because otherwise would have more than two continuations.
According to the entry ring of exponent (http://planetmath.org/RingOfExponent), any non-zero element of the field is uniquely in the form
with a unit of and integers. The both continuations and of the triadic exponent are then determined as follows:
|Title||determining the continuations of exponent|
|Date of creation||2013-03-22 18:00:16|
|Last modified on||2013-03-22 18:00:16|
|Last modified by||pahio (2872)|