By Alexey L. Gorodentsev (auth.)
This publication is the 1st quantity of a thorough “Russian-style” two-year graduate direction in summary algebra, and introduces readers to the elemental algebraic buildings – fields, earrings, modules, algebras, teams, and different types – and explains the most rules of and techniques for operating with them.
The direction covers colossal parts of complex combinatorics, geometry, linear and multilinear algebra, illustration conception, type idea, commutative algebra, Galois thought, and algebraic geometry – subject matters which are usually ignored in ordinary undergraduate courses.
This textbook relies on classes the writer has carried out on the self reliant collage of Moscow and on the college of arithmetic within the better university of Economics. the most content material is complemented by way of a wealth of workouts for sophistication dialogue, a few of which come with reviews and tricks, in addition to difficulties for self sustaining study.
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Extra resources for Algebra I: Textbook for Students of Mathematics
14 The sides of a regular planar n-gon lying in three-dimensional space are painted in n fixed different colors, one color per side, in all possible ways. How many different painted n-gons do we get if two colored n-gons are considered the same if one can be obtained from the other by some motion in three-space? 13 1 m2 The total degree of the monomial xm 1 x2 n xm n equals Pn iD1 mi . 15 How many different necklaces can be made from 5 red, 7 blue, and 11 white otherwise identical glass beads? 16 All the faces of a regular (a) cube, (b) tetrahedron, are painted using six fixed colors (different faces in distinct colors) in all possible ways.
F C g/ D f C g : 2 3 See Sect.
N/ are also called primitive roots (or primitive residues) modulo n. n/. (a) Prove the existence of a primitive residue modulo p for every prime p 2 N. n/ have pairwise coprime orders k1 ; k2 ; : : : ; kn . Find the order of the product a1 an . k; m/. (d) Fix a prime p > 2 and a primitive residue % modulo p. modp2 / ; (3) the class Œ% C p# is a primitive residue class modulo pk for all k 2 N. (e) For k 2 N and prime p > 2, prove the existence of a primitive residue modulo 2pk . (f) Is there a primitive residue modulo 21?
Algebra I: Textbook for Students of Mathematics by Alexey L. Gorodentsev (auth.)