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The original Galois theory was developed to answer the question of the expressibility of the roots of polynomial equations with arithmetic operations and radicals.

However it seems that later presentations of the theory seem to drop the reference to equations and instead seem to focus on fields and their extensions.

What motivated this change of emphasis?

(If this topic has been discussed anywhere else, references would be most welcome)

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Like any theory, Galois theory matured with time. It was understood that the theory can be expressed more clearly in terms of fields and many applications outside of the original confines the theory was designed for emerged. This is also part of a general trend in modern mathematics, namely abstracting and axiomatizing.

This is not unique to Galois theory. Linear algebra was originally the study of systems of linear equations, but today we concentrate on the concept of an abstract vector space. Analysis used to be very much concerned with the study of certain metric spaces, e.g., spaces of functions, but the concept of an abstract metric space is, today, central. Measure theory was designed to produce a better behaved integral than the Riemann integral, but today the concept of an abstract measure space is fundamental.

Basically any new theory is designed to answer some question and thus is fraught with specific details that can, and should, be abstracted away so that we can see the forest and not just trees.

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  • $\begingroup$ This is a very interesting reply. Are you perhaps familiar with any book on the development of these theories from 19th century onwards? $\endgroup$ – Timotej Nov 27 '13 at 13:40
  • $\begingroup$ @Timotej, sorry but I don't know of any historic account of these developments. You can try to read some of the original works, but I have no idea how accessible they are. $\endgroup$ – Ittay Weiss Nov 27 '13 at 13:51
  • $\begingroup$ Thanks. If you don't mind another question - do you perhaps know of any other application of Galois theory apart from solubility of equations by radicals? $\endgroup$ – Timotej Nov 27 '13 at 13:57

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