Among several algebraic structures that appear in mathematics, groups are the simplest one. Originally groups consisted of only transformation groups and later in 19th century groups appeared in the context of theory of algebraic equations. Later, to be exact in 1882, these were generalized to abstract groups defined by a set of axioms.

Isomorphism gives a method of comparing group structures defined on different sets. Homomorphism between two groups is a more general concept. We introduce the concept of a normal subgroup which plays a very important role in group theory.

Cayley's theorem states that every group is isomorphic to the group of permutations. The consequences of this theorem are very important. It means that, for the study of groups, it is enough to study the permutations groups. In the case of finite groups it is enough to study the symmetric group Sn and its subgroups.

The conjugacy relation is an equivalence relation. By decomposing the group into equivalence classes, called the congruence classes, we obtain the class equation for G which is an important and useful tool. We determine the class equation for Sn by determining the conjugacy classes in terms of the cycle structures of the permutations.

Ring is an algebraic system having two binary operations subject to axioms similar to the properties satisfied by the usual addition and multiplication in Z. In this Unit we give the axiomatic definition of a ring and study some of its elementary properties.

The fundamental theorem of arithmetic states that every non-zero integer can be uniquely expressed (except for the sign) as a product of powers of primes. In this Unit, we shall study rings where this property holds. We formulate the concepts like divisibility, factorization, prime elements, irreducible elements, greatest common divisor, etc., in any integral domain R.

Recall that normal subgroups are special kind of subgroups which are kernels of group homomorphisms. In the case of rings, its role is played by a special kind of sub rings called the ideals. For any subring S of the ring R it is not always possible to have a ring structure on R/ S. However this can be done when the subring is an ideal.

In § 2.2.4 we discussed the polynomial ring R[X] in the indeterminate X with coefficients from the ring R. The nature of the ring R[X] depends very much on the coefficient ring R itself. For example, R[X] is an integral domain whenever R is an integral domain (Proposition 2.2.4.1). When R is a field, we show that R[X] is a Euclidean domain.

Abstract algebra has three basic components: groups, rings and fields. Fields play a central role in algebra. Results in field theory find important applications in algebraic number theory. Field theory happens to be the language in which a number of classical problems such as the Greek's problem of ruler and compass constructions were rephrased and solved.

Given any algebraic structure such as group, ring or a vector space, we have considered sub-algebraic structures such as subgroups, sub rings or a subspace. In a similar fashion we shall define subfield and an extension field of a field. In group theory it is customary to ask questions about the subgroups of a given group.

We have observed that in general we have to go to a bigger field. namely a field K containing F to get the roots of a polynomial defined over F. Moreover, the nature of roots, as to whether they are simple or multiple has to be studied. We shall do a systematic study of such questions by studying field extensions.

Finite fields is one of the most beautiful and important area of abstract algebra, It was introduced by Galois in 1830 in his proof of the insolvability of the general quintic equation. To this day, matrix groups over finite fields are among the most important class of groups.

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