Is u Algebraic Over K if u^2 is Algebraic Over F?

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SUMMARY

The discussion centers on proving that if \( u \in K \) and \( u^2 \) is algebraic over a field \( F \), then \( u \) is algebraic over the field extension \( K \). Participants clarify that since \( u \) is an element of \( K \), it is inherently algebraic over \( K \). The key insight is that if \( u^2 \) is algebraic over \( F \), one can utilize the polynomial \( f(x^2) \) to demonstrate the algebraicity of \( u \) over \( F \). This realization simplifies the proof process significantly.

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jeffreydk
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I'm currently trying to prove that (for a field extension [itex]K[/itex] of the field [itex]F[/itex]) if [itex]u\in K[/itex] and [itex]u^2[/itex] is algebraic over [itex]F[/itex] then [itex]u[/itex] is algebraic over [itex]K[/itex].

I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this would be greatly appreciated.
 
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Well clearly u is algebraic over K (it is an elemnt of K) so I'm guessing you mean to say that u is algebraic over F. Well if u^2 is algebraic over F then let f be a polynomial in F[x] such that f(u^2)=0. I don't want to give it away but if you think about the polynomial f(x^2)...
 
Oh wow yea, ok it's pretty clear. I think I was complicating things. Thanks.
 

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