A question on field of fractions and the Laurent series.?

Question

If F is a field, prove that the field of fractions of F[[x]](the ring of formal power series in the indeterminate x with coefficients in F) is the ring F((x)) of formal Laurent series. Show the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)). Consider the series for e^x.

Here the Laurent series is of the form $\sum_{n\geq N}^\infty a_k x^k$ where N is an integer.

Answer 1

First show that the reciprocal of every element of F[[x]] can be written as a Laurent series: this is best done by inductively defining the coefficients of the quotient. Conversely, show that the reciprocal of every Laurent series can be expressed as a Laurent series. This shows that F((x)) is a field (assuming you check the obvious properties) and contains the field of fractions of F[[x]]. Since a sufficiently high power of x can be multiplied by a Laurent series to get an element of F[[x]], this completes the proof that F((x))) is the field of fractions of F[[x]].

As for Z[[x]]. The idea is to show that the power series for e^x (which is clearly in Q[[x]]) is not of the form f(x)/g(x) for f,g in Z[[x]]. But if f/g is in lowest terms, it is easy to see that g must be a constant (otherwise there would be terms of the expansion with negative exponents), i.e. g is in Z. But the expression for e^x cannot be so written!

Answer 2

Field Of Fractions

Answer 3

Hey, you forgot to change your question so that it was not in Tex.