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How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of something like this.

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have you read the wikipedia article? – Grigory M Aug 13 '10 at 18:16
@Grigory M: Yes – anonymous Aug 13 '10 at 18:50
up vote 3 down vote accepted

I'm certainly no historian, so let me give my impression of the history in a CW answer and others can improve on this answer.

Groups were understood as permutations of objects long before they were understood as abstract groups, just as manifolds were understood as nice subsets of $\mathbb{R}^n$ long before they were understood as abstract manifolds. Most notably, Galois worked with permutations of the roots of a polynomial in order to understand the solvability of polynomials by radicals. Similarly, number theorists like Kummer worked with concrete examples of number rings and fields.

Some of the people responsible for the spread of the abstract point of view are (in no particular order) Hilbert, Noether, Artin, van der Waerden, (Dedekind?), .... together they more or less determined the abstract algebra curriculum as we now understand it. Their reasons for doing everything axiomatically include that the proofs are shorter and more powerful this way; in particular, algebraic geometry needed rigorous tools and commutative algebra provided them (and Zariski used them).

One big payoff of the abstract point of view is that people now recognized fundamental groups and homology groups in topology as groups. Before the abstract study of groups these objects, while defined in some sense, weren't recognized as groups because they didn't permute anything.

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Don't forget Cayley: he's the one who went from "groups are collections of permutations" to try the more abstract route. Dedekind set the groundwork for the abstract notion of "ring". – Arturo Magidin Aug 13 '10 at 18:27
By the way, just to give some sense of the timeline, "long before" (in the discussion of the passage from groups of permutations to abstract groups) is not so long: Galois's work dates from about 1830, and Cayley's from the 1850s. (It's not clear that Cayley's abstract viewpoint caught on immediately, since permutation groups still dominated the subject for some time afterwords. But already Lie was defining Lie groups in the later 1800s, and Klein was stating the Erlangen program around a similar time.) – Matt E Aug 13 '10 at 21:36

Let me address the question: "what made Mathematicians ... think of something like this"?

The answer is: Galois, in studying the problem of factorization of polynomials, realized that reasoning about the symmetries of the roots could be more powerful a tool than studying explicit formulas (which, loosely speaking, had been the basic method in the theory of equations up to that time).

As this structural/conceptual point of view began to reveal its power in solving difficult concrete problems, more and more mathematicians began to think in this way. The structural concepts were then isolated from their concrete settings, and this is how they are taught today. But the motivation was, and for many remains, the applications of these abstract notions to concrete problems. (A standard but helpful example is the pivotal role that group theory, Galois theory, cohomology, and ring theory played in the proof of Fermat's Last Theorem.)

Furthermore, nowadays we train ourselves and our students to recognize any hint of structure in a problem, and to exploit such structure to the hilt. So these concepts have become basic tools in the problem-solving toolkits of contemporary mathematicians.

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With respect to the introduction of fields, Erich Reck in his "Dedekind's Contributions to the Foundations of Mathematics", cites Dedekind as being the first to introduce this concept into mathematics, in the context of his work in algebraic number theory. Reck's article goes into some detail about Dedekind's other contributions to mathematics that seem like they may be of interest, for example, his work on lattices.

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