Some applications of topology to algebra (Q1897196)

The paper under review appeared firstly in the year 1952 in [Univ. Politec. Torino, Rend. Sem. Mat. 11, 75-91 (1952; Zbl 0049.30603)]. But it is still interesting and has some actuality. The oldest example of a topological method in proving an algebraic theorem is the case of real polynomials of odd degree; they have (at least) one real zero. The author presents a two-dimensional (via theorem of Rouché) and an \(n\)-dimensional generalization (via Brouwer's topological degree of a mapping). He then proves via Brouwer's fixed point theorem the theorem of Frobenius, that every non-negative \(n \times n\)-matrix has (at least) one non-negative eigenvalue. Furthermore he proves some other theorems in linear algebra (matrices) and algebra (quaternions, division algebras) by topological methods and states some questions which are in the meantime partially answered (e.g. by Milnor on tangent bundles on spheres).