height of an algebraic numberHeightOfAnAlgebraicNumber22003-01-31 15:34:122004-04-12 16:46:59Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereSuppose we have an algebraic number such that the polynomial of smallest degree it is a root of (with the co-efficients relatively prime) is given by:
$$
\sum_{i=0}^n a_i x^i .
$$
Then the height $h$ of the algebraic number is given by:
$$
h = n + \sum_{i=0}^n |a_i| .
$$
This is a quantity which is used in the proof of the existence of transcendental numbers.
\begin{thebibliography}{99}
\bibitem{shaw} Shaw, R. Mathematics Society Notes, 1st edition. King's School Chester, 2003.
\bibitem{stewart} Stewart, I. Galois Theory, 3rd edition. Chapman and Hall, 2003.
\bibitem{baker} Baker, A. Transcendental Number Theory, 1st edition. Cambridge University Press, 1975.
\end{thebibliography}