What do you mean by "rational power series"? Rational coefficients?
Given an element having a property that no non-constant polynomial in has as a zero, we call it "transcendental". For example, is an example.
However, if we allow "infinite polynomials", i.e. power series then it is no longer transcendental. Because is a zero of .
My question is: Is there an element in which is purely transcendental? Meaning it has no rational power series even.
Correct me if I'm wrong, but we should be able to represent any real number as a decimal, yes? Which means that all real numbers, including the transcendentals can be expressed as for some choice of . (This argument would seem to depend on the Axiom of Choice?) So my stab at the answer is that there is no such transcendental number.
-Dan
I wonder where he comes up with those
Ok, suppose such a number x exists -- not a root of any power series with rational coefficients. We will prove Hacker's questions are transcedental.
The set S={1,x,x^2,...} is then linearly independent over Q, and thus constitutes a transcedence basis for R (over Q). But S is countable, while the degree of the extension of R over Q has the power of the continuum.
"Quod erat demonstratum"
It is obvious that the set of numbers which are roots of some rational power series has the same cardinality as , but that does not prove tht the set exhausts .
It is also obvious tha the set is dense in , as it inherits this propety from the set of algebraic numbers which it contains.
RonL
How do we know (...) spans?
It is also obvious tha the set is dense in , as it inherits this propety from the set of algebraic numbers which it contains.
You are both right, I mistakingly took it for a maximal set.