# height of an algebraic number

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.

## References

• 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.
Title height of an algebraic number HeightOfAnAlgebraicNumber 2013-03-22 13:24:34 2013-03-22 13:24:34 kidburla2003 (1480) kidburla2003 (1480) 17 kidburla2003 (1480) Definition msc 03E10 height AlgebraicNumbersAreCountable