Yes and no, it's a bit too simplistic and doesn't explain the actual "why" of Galois Theory, just the how. The brilliant insight Galois figured out is that there is a fundamental connection between fields and groups, but that is just the "technique" with which he solved the problem. The "why even bother" is a bit more complex but simply put Galois wanted to establish a criterion to determine what polynomials are solvable or unsolvable in which fields. E.g. we know x^2=-1 is solvable in C with x=i but not in real numbers. Can we generalize that proof to such a degree that we can mechanistically run it for arbitrary polynomials in arbitrary fields?

What it states is correct and it gives you a good overview over what you do in a Galois theory course. It does, however, not give you an idea of why this is interesting. When just reading that article one might get the idea that some mathematicians just had too much free time.

I tried to motivate the questions leading to Galois Theory in https://news.ycombinator.com/item?id=41258726 in a way that is hopefully accessible to more down-to-earth programmers and engineers.

I should probably add why I think the motivation is so important here. For pure engineers, numbers are a tool. They ask: What can I build with numbers? Pure mathematicians ask a different question. They are interested in the limits of numbers. They ask: What can I not build with numbers? Studying these two questions is deeply related but also a constant source of frustration for engineers taking math courses designed for mathematicians by mathematicians.

Galois theory, is a theory of "no". It ultimately serves to answer several "Can I build this?" questions with no. This makes it very interesting to pure mathematicians. However, for pure engineers that are looking for numeric machine parts that can be assembled in other useful ways to actually build something... Galois theory can be quite disappointing.

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