I found the Jacobi symbols to be about the same as the Legendre symbol. However, I did not quite understand the distiction. I take it that the Jacobi symbol is just a generalization, but I do not quite understand what it is generalizing to.
From this, I think that we will just keep generalizing things to more and more abstract areas. I suppose it give more power in definining things, but sometimes, it is hard to see the big picture.
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