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I'm curious as to how groups, rings, fields got their names. Did someone just start calling these structures by those names, or is there a (not entirely) arbitrary reason for them?

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See this question and this question on math.SE, and this question on MathOverflow (and this question and this question for other terminology). – Zev Chonoles Jul 2 '12 at 0:27
I think Galois used the word group to refer to permutation groups of zeros of polynomials. I think Arthur Cayley is the author of the modern definition, but he was probably just using established terminology rather than endorsing the idea that that's the best possible word. – Michael Hardy Jul 2 '12 at 1:13

You should look up Evariste Galois and Niels Abel. Both were incredibly talented Mathematicians who died at very early ages (around their early twenties). I know that concerning Galois, he died in a very mysterious way involving a duel over a woman. Personally, both are some of my heroes in Mathematics.

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