**Carl Ludwig Siegel (1896-1981)**

**I mention only a small fraction of Siegel's achievement, and in the process am omiting many remarkable results.**

**Siegel's (1917) Theorem**
**. Let **
** be any real algebraic number of degree **
**n**
** (**
**at least**
** 3), and let **
** be any positive constant (**
**however small**
**), then there is a positive constant **
** (i.e. the value of **
**c**
** depends **

**only**
** on **
** and **
**) such that**

** **
**>**
** **
** ... (i'')**

**and Siegel **
**conjectured**
** that the above inequality could be **
**improved**
** to**

** **
**>**
** **
** **

**which, if true, would, of course, be best possible (in a sense), since by a classic (and easy to prove theorem of Dirichlet) one has: for every real irrational number there are infinitely many distinct rational numbers **
** (**
**p**
** and **
**q**
** integers, with **
**) such that**

**Siegel was the first to prove transcendence results involving elliptic functions. **

**Siegel's (1932) Theorem**
**. Let P(**
**z**
**) be the Weirstrass elliptic function with algebraic number **
**invariants**
** **
** and **
** in the equation **
** (the standard connection between P(**
**z**
**) and its derivative P '(**
**z**
**)), then at least one of any **
**fundamental**
** pair of periods of P(**
**z**
**) is transcendental. **

**At the end of chapter 3 of his book **
**Transcendental Numbers**
**, the one dealing with Hilbert's seventh problem, Siegel wrote (to illustrate the state of ignorance, I have slightly changed Siegel's notation and wording, and made some emphases): **
**The result on the transcendency of **
** can also be stated this way: If **
** **
**are algebraic numbers, **
**, **
**, **
**then the ratio **
** **
**is either rational or transcendental. In other words, the logarithm of any algebraic number relative to any algebraic base is either rational or transcendental. However...it is **
**not even known**
** whether there **
**cannot**
** exist an inhomogeneous linear relation **
** **
**with**
** **
**quadratic irrational**
** **
**.**

** And that question of Siegel's concerned only two logarithms, with severely restricted (though completely non-trivial) quadratic algebraic **
**. What a later triumph it was for Alan Baker to completely settle, not just that question of Siegel, but the completely general version of it: **
**n**
** logarithms, and algebraic **
**s. I will come to that in the later Baker section.**

** At the end of that same chapter Siegel also remarks: **
**Another example showing the narrow limits of our knowledge on transcendental numbers is the following one: Since e and **
** **
**are both transcendental, **
__not both__
** numbers **
** **
**and**
** **
** **
**can be algebraic **
**[that, by the way, is simply a particular case of a completely general remark, namely: if **
** are both transcendental, then **
** and **
** cannot both be algebraic]**
**; but we do not even know whether **
** **
**or**
** **
** **
**are irrational.**