# 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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## closed as not a real question by Ross Millikan, Adrián Barquero, mixedmath♦, t.b., Chris EagleJul 21 '11 at 10:36

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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