The Gelfond-Schneider Theorem states that if a and b are algebraic numbers, a is not equal to zero or one, and b is irrational, then a raised to the power b is transcendental. Proved independently by Aleksandr Gelfond and Theodor Schneider in 1934, it resolved Hilbert's seventh problem and gives a general method for proving numbers such as two raised to the power of the square root of two are transcendental.
Facts
StatementThe Gelfond-Schneider theorem establishes the transcendence of a large class of numbers: if a and b are algebraic numbers with a not equal to 0 or 1 and b irrational, then any value of a^b is transcendental. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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In Branch
Proved By
Source Gelfond-Schneider Theorem (Wikipedia)
Source Gelfond-Schneider Theorem (Wikipedia)
Sources
1. Gelfond-Schneider Theorem (Wikipedia)
Wikimedia FoundationLead paragraph
In mathematics, the Gelfond-Schneider theorem establishes the transcendence of a large class of numbers.
History section
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Proved By: Alexander Gelfond, Lead
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Proved By: Theodor Schneider, Lead
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
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