Posted: Wed Oct 20, 2004 9:46 pm    Post subject: How to use algebra to prove it?

I saw the following questions in Topics in Algebra. Can you use algebra to prove it?

1. If is an odd prime and if where and are integers, prove that . If , prove that .

2. Prove that , where (This should be easier)

Posted: Thu Oct 21, 2004 11:12 pm    Post subject:

I think the idea is that, in the field , is just a rearrangement of . These numbers sum up to be , which is a multiple of p. You can make it rigorous.

The second question should be an application of Lagrange theorem. The reduced residue system modulo is a multiplicative group of order , and is a subgroup of it.

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世上沒有完美的人完美的事，而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來，與時代一起進步去追求無止境的完美。