hilbert's seventh problem

Hilbert's seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). In its geometric formulation, it asks whether the following statement is provably true:

In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, then the ratio between base and side is always transcendental.

A special case of this problem asks:

Is a^b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?

When b is rational, a^b will be algebraic. The special problem was solved by Aleksandr Gelfond in 1934, and refined by Theodor Schneider (1911 - ) in 1935. They proved that a^b is transcendental when b is both algebraic and irrational. This result is known as Gelfond's theorem or the Gelfond-Schneider theorem. From the point of view of generalisations, this is the case

blog (α) + log(β) = 0

of the general linear form in logarithms. See also:

<< Previous

Word Browser

Next >>

alexander bustamante
tubal uriah butler
cerinthus
leopold ii
beast (comics)
stephen gray
mile lahoud
letsie iii of lesotho
peter maxwell davies
offset
hugh i of jerusalem

list of ambassadors and high commissioners to the united kingdom
lvaro obregn
power supply in norway
luther blissett (nom de plume)
goku junior
monarchs of the armenian kingdom of cilicia
nauruan indigenous religion
mohammed jamal khalifa
dennis eckersley
aston sandford
terry nichols

geneva accord
vladimir zhirinovsky
seijogakuen mae station
baburam bhattarai
giovanni angelo montorsoli
plutarco elas calles
haddenham, buckinghamshire
nadi
haddenham
dan o'neill
heinkel

bethany hamilton
commonwealth jack
protectorate jack
brush tailed rock wallaby
grand calumet river
relaxation
presidential standard (ireland)
airman
todd haynes
commie awards
new queer cinema