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I'm sorry in advance if here is not the suitable place to ask this question, and people can feel free to vote to close this if that's the case. However, since I'm not sure about this issue, I'll ask it in any way:
I was reading Hungerford's Algebra, and my friend working on food science saw the title "Fundamental Theorem of Galois Theory", and asked me what it's about. (Seeing the word "fundamental", he thought it should be a very basic, simple concept.) Then, I said something like "there are structures called groups, fields, and they have some special subsets called subgroup, subfields. In order to understand subfields, mathematicans makes a correspondence between subgroups of some associated group, which is easier to work with and better understood, compared to fields."
He said it made some sense, but I'm sure it was a terribly bad explanation. My question is how do people generally deal with this issue? Is this only my problem (which really might be the case), or is this a general problem with pure math people?
This issue really annoys me in the sense that, for example, when you consider some person working on food science, psychology, chemistry etc., they're generally able to tell roughly what they're trying to do. But, I'm always having very hard time when I'm asked the same question. I'm a PhD student now, and I plan on spending whole my life with mathematics. But, I'm afraid that, say in my 50 (if I live that much), if someone asks me what I did in my entire 50 years, and I fail to answer, that would be a shameful situation for me, I believe.
I tried my best to express my question in a good way, but still I'm sorry if it's not good enough, and don't hesitate to let me know if I should write some parts more clearly/explicitly.