# Elliptic curves and p-Adic linear independence

1. Introduction The problem of finding roots of a given polynomial is always a natural and big question in mathematics. It is well-known that every polynomial with complex coefficients of positive degree has all roots in the complex field C and in particular, so does every polynomial with rational coefficients of positive degree. Dually, to study the arithemetic of complex numbers, that is given α ∈ C, one may naturally ask whether there is a non-zero polynomial P in one variable with rational coefficients such that P (α) = 0? If there exists such a P we call α algebraic, otherwise we call α (complex) transcendental. The most prominent examples of transcendental numbers are e (proved by C. Hermite in 1873) and π (proved by F. Lindemann in 1882). Apart from the complex field C, there is another important field, the so-called (complex) p-adic number field (first described by K. Hensel in 1897) for each prime number p. Namely, it is a p-adic analogue of C which is denoted by Cp. Note that by construction, Cp is an algebraically closed field containing Q, therefore one can analogously give the definition of p-adic transcendental numbers as follows. An element α ∈ C p is called (p-adic) transcendental if P (α) ̸= 0 for any non-zero polynomial P (T ) ∈ Q[T ]. Transcendence theory in both domains C and Cp has been studied and developed by many authors. In order to investigate the theory more deeply, one can naturally put the problem in the context of linear independence. For instance, if α is a number (in C or Cp) such that 1 and α are linearly independent over Q, then α must be transcendental. Indeed, it follows from the trivial equality: α · 1 − 1 · α = 0. One of the most celebrated results in this direction is due to A. Baker. Namely, in 1967 he proved the following theorem (see [1])

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