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Quotient Rings, Laurent Series and Power Series

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

Ps. Here F[[x]] is the ring of formal power series in the indeterminate x with coefficients in F.

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Solution Summary

Quotient rings, Laurent series and power series are investigated and well presented in the solution.

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Proof:
First, I claim that the field of fractions of F[[x]] is F[[x]] itself.
Since F[[x]] is the ring of formal power series, then without considering
convergence, we have 1/x = 1/(1-(1-x))=1+(1-x)+(1-x)^2+... and it belongs
to F[[x]]. This implies that (1/x)^k also belongs to F[[x]]. Next, we
consider any f(x) in F[[x]], ...

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