Suppose 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.

Bibliography

1

Shaw, R. Mathematics Society Notes, 1st edition. King's School Chester, 2003.

2

Stewart, I. Galois Theory, 3rd edition. Chapman and Hall, 2003.

3

Baker, A. Transcendental Number Theory, 1st edition. Cambridge University Press, 1975.