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Menu Skip to primary content Home About Search Matt Baker’s Math Blog Thoughts on number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics Algebraic Values of Transcendental Functions at Algebraic Points In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem , which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebraic values at algebraic points? The connection with is that Lindemann proved in 1882 that the transcendental function takes transcendental values at every nonzero algebraic number. Since by Euler’s formula, this proves that , and hence itself, is transcendental. In light of this theorem, it is natural to wonder what if anything is special here about the function and the point . Ferdinand von Lindemann One thing that’s special about is that if is algebraic and is also algebraic, then both and are algebraic for all , and these numbers are all distinct . So one

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