Transcendental number

2020 Mathematics Subject Classification: Primary: 11J81 [MSN][ZBL]

A number that is not a root of any polynomial with integer coefficients. The domain of definition of such numbers is the field of either the real, complex or $p$-adic numbers. The existence and explicit construction of transcendental numbers was provided by J. Liouville [1] on the basis of the following fact, noted by him. Irrational algebraic numbers do not have "very good" approximations by rational numbers (see Liouville theorems). Similar considerations enable one to construct $p$-adic transcendental numbers. G. Cantor [2], after discovering the countability of the set of all algebraic numbers and the uncountability of the set of all real numbers, thus proved that the transcendental real numbers form a set of the cardinality of the continuum. E. Borel [3], after introducing the first concepts of measure theory, established that "almost all" real numbers are transcendental. It was later found that Liouville transcendental numbers form an everywhere-dense subset of the real axis, having the cardinality of the continuum and zero Lebesgue measure. Despite the fact that already in the middle of the 18th century there arose the conjecture on the transcendency of numbers such as $e$, $\pi$, $\log 2$, $2^{\sqrt2}$, etc., proofs of this could not be obtained. The transcendency of $e$ was proved by Ch. Hermite [4], that of $\pi$ and, more generally, of logarithms of algebraic numbers by C.L.F. Lindemann [5], that of $2^{\sqrt2}$ by A.O. Gel'fond [6]; C.L. Siegel [7] developed a general method for proving transcendency and algebraic independence of the values at algebraic points of entire functions of a specific class (the $E$-functions), satisfying a linear differential equation with polynomial coefficients (cf. Siegel method). Gel'fond [8] and T. Schneider [9] simultaneously and independently proved that $\alpha^\beta$ is transcendental if $\alpha \ne 0,1$ is algebraic and $\beta$ is an algebraic irrational (the so-called Hilbert's seventh problem); A. Baker [10] proved the transcendency of products of numbers of the form $\alpha^\beta$ under natural restrictions. Similar results have been obtained for $p$-adic transcendental numbers (including Engel's theory of $E$-functions). The development of methods of the theory of transcendental numbers has proved to have a strong influence on new studies in Diophantine equations [10], [11].

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The results of Gel'fond and Schneider imply that for any $\alpha,\beta \in \bar{\mathbf{Q}} \setminus \{0,1\}$, if $\log\alpha/\log\beta \not\in \mathbf{Q}$ then $\log\alpha/\log\beta \not\in \bar{\mathbf{Q}}$. Baker's generalization asserts that for any $\alpha_1,\ldots,\alpha_n \in \bar{\mathbf{Q}}$, linear independence of $\log\alpha_1,\ldots,\log\alpha_n$ over $\mathbf{Q}$ implies linear independence over $\bar{\mathbf{Q}}$. Moreover, one can give effective lower bounds for such linear forms in logarithms. This has profound consequences for the theory of Diophantine equations (see [10]). Gel'fond's and Schneider's method has been further generalized to include $\bar{\mathbf{Q}}$-linear independence of periods and quasi-periods of elliptic curves (see [b1]) and finally, through the work of G. Wüstholz, P. Philippon and M. Waldschmidt, this has resulted into very general statements of $\mathbf{Q}$-linear independence on commutative algebraic groups defined over $\mathbf{Q}$.

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