By V. B. Alekseev

Translated via Sujit Nair

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Definition 23 If on a certain set two binary operations (addition and multiplication) are defined which possesses the above three properties, then this set is called a field. Problem-197 Are the following subsets of real numbers with the usual operations of addition and multiplication fields ? a) all natural numbers; b) all integers; c) all rational numbers; g) all numbers √ of form r1 + r2 2, where r1 and r2 are arbitrary rational numbers. Problem-198 Prove that in any field a · 0 = 0 · a = 0 for any element a.

Problem-188 Prove that for n ≥ 4, An is noncommutative. Problem-189 Prove that the alternating group An is a normal subgroup of the symmetric group Sn and to build the decomposition of the group Sn by An . Problem-190 To determine the number of elements in the group An . Problem-191 Prove that the groups S2 , S3 and S4 are solvable. We now prove that the alternating group A5 is non-solvable. One of the proofs consists of the following. Inscribe five tetrahedrons labelled by 1, 2, 3, 4, 5 in a dodecahedron in such a way that to every 36 2 Groups rotation of the dodecahedron there corresponds an even permutation of the tetrahedra and different rotations correspond to different permutations.

7 Continuous curves 53 1 O 1 O x x -2i -2i Fig. 6. O 1 O 1 Fig. 7. Fig. 8.