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Is there any easy nice example to say that this element of $k((t))$ is transcendental over $k(t)$ (We can use the cardinality argument for the existence of transcendental element.But, I am looking example). Thank you

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$k((t))$ denotes formal power series? What about $\phi=\prod_n \frac1{1-t^n}$? It has infinitely many "poles" in $\bar k$, hence in any polynomial with coefficients in $k(t)$ there is some pole $z$ not occuring as pole or zero of any coefficient. Therefore $z$ is also a pole of order $\deg p$ of $p(\phi)$. –  Hagen von Eitzen Sep 16 '12 at 15:17
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