The Cubic Formula and Galois Theory

I read a couple of books fairly recently that discuss the cubic equation: Barry Mazur’s Imagining Numbers: (particularly the square root of minus fifteen) and William Dunham’s Journey through Genius. They got me interested in this formula, which had always been an afterthought (to the extent to which it had been a thought at all) in my mathematics education.

The cubic and quartic formulas were perhaps the greatest accomplishments of Renaissance mathematics, and they were a key spur to Evariste Galois’ beautiful study of the symmetry inherent in polynomial equations. Yet I got a PhD in mathematics without knowing much more than the bare fact that such formulas exist. It should certainly be possible to understand these formulas in light of Galois theory, but such explanations are not easy to find.

I wrote this paper to pass on what I learned about the cubic equation, and how it can be derived in the context of Galois theory.

It helped me understand the cubic formula, and it helped me understand Galois theory better by studying a semi-concrete example. The paper should be accessible to an undergraduate who has had a course in Galois theory; in any case, it contains a review of the necessary Galois theory, and an adventuresome reader who has not had such a course but is familiar with the basics of groups, fields, and linear algebra might get enough out of it to want to study Galois theory in more depth.

If you read it, I hope you enjoy it. Comments, questions, suggestions, etc. are welcome here.

Edit Jan 15, 2019: I edited the original version slightly to include a couple of diagrams.

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