ring of exponent
of is called the . It is, naturally, an integral domain. Its elements are called .
Theorem 2. The ring only one prime element , when one does not regard associated elements as different. Any non-zero element can be represented uniquely with a in the form
where is a unit of and . This means that is a UFD.
Remark 1. The prime elements of the ring are characterised by the equation and the units the equation .
Remark 2. In an algebraically closed field , there are no exponents (http://planetmath.org/ExponentValuation). In fact, if there were an exponent of and if were a prime element of the ring of the exponent, then, since the equation would have a root (http://planetmath.org/Equation) in , we would obtain ; this is however impossible, because an exponent attains only integer values.
Theorem 3. Let be the rings of the different exponent valuations of the field . Then also the intersection
is a subring of with unique factorisation (http://planetmath.org/UFD). To be precise, any non-zero element of may be uniquely represented in the form
|Title||ring of exponent|
|Date of creation||2013-03-22 17:59:43|
|Last modified on||2013-03-22 17:59:43|
|Last modified by||pahio (2872)|
|Defines||ring of an exponent|
|Defines||ring of the exponent|
|Defines||integral with respect to an exponent|