Question in fields

[chk2]

hi!
I have a problem with the following question. I translated it to English so I hope I didnt make a mistake..

there is a fild F. We assume that there is no "a" that makes (a^2)+1=0.

show that if there are X and Y in F , and (X^2)+(Y^2)=0 so X=Y=0 .



thanks!!!

 

[JeffM]

hi!
I have a problem with the following question. I translated it to English so I hope I didnt make a mistake..

there is a fild F. We assume that there is no "a" that makes (a^2)+1=0.

show that if there are X and Y in F , and (X^2)+(Y^2)=0 so X=Y=0 .



thanks!!!

Helping with proofs is tricky because we do not know what theorems you are allowed to use

Here are some hints

\(\displaystyle Given: [1]\ F\ is\ a\ field;\)

\(\displaystyle [2]\ a \in F \implies a^2 + 1 \ne 0;\)

\(\displaystyle [3]\ x \in F;\)

\(\displaystyle [4]\ y \in F;\ and\)

\(\displaystyle [5]\ x^2 + y^2 = 0.\)

\(\displaystyle y^2 \in F.\)

\(\displaystyle y \ne 0 \implies \left(y^2\right)^{-1} \in F.\)

Now assume \(\displaystyle y \ne 0\) and try a proof by contradiction. Do you see it now?

 

[chk2]

thanks!!!!!!!

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