# the algebraic elements form a field [closed]

I proved this simple thing, but using some simple field theory. I want to know whether I can prove it with simpler tools. The proof is not difficult, it uses only a little field theory, like the idea of extensions of finite degree. But can I prove it only using the definitions?

Another related question: If I know that the element $u$ satisfies some polynomial equation $p(x)=0$, can I find the polynomial for $1/u$ only knowing $p(x)$?

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You've asked 8 questions, and accepted no answers. Why ask a 9th, when clearly you aren't satisfied with the answers you get here? –  Gerry Myerson Jul 21 '11 at 4:59
Hint: laimonylop –  André Nicolas Jul 21 '11 at 5:07
@André Nicolas: I put your hint into a well-known search engine before I thought to reverse it. It is amazing how many hits there are with varying numbers of capitals. –  Ross Millikan Jul 21 '11 at 5:18
@André Nicolas Nice hint. I couldn't understand it at first but now it all makes sense =P –  Adrián Barquero Jul 21 '11 at 5:22
@Adrian: no, it makes esnes. –  gary Jul 21 '11 at 6:39