[FOM] Jan Pax's question on Ring
smohan at isical.ac.in
smohan at isical.ac.in
Fri Aug 2 01:46:23 EDT 2013
Response to Jan Pax's question on rings:
``If I have understood the question correctly, the answer is no even if
$R$ is a field:
The field of rational functions $R(X)$ is neither contained in nor
contains the ring of power series $R[[X]]$.
It is clear that $X$ has no inverse in in $R[[X]]$. So, $1/X$ is in $R(X)$
(the field of rational functions in $X$ over $R$) but has no power series
expansion.
The exponential power series $\sum X^n/n!$ does not belong to any
extension field of $R$ of finite transcendence degree but $K$ is of
transcendence degree 1. So, the exponential power series does not belong
to $R(X)$.
''
SHASHI M. SRIVASTAVA
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