Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $K\supseteq F$ be a field extension and $x,y\in F$. Can someone please explain to why the equivalence $$ x,y\text{ algebraic over }F\ \Leftrightarrow\ x+y\text{ and }xy\text{ are algebraic over }F $$ should hold ?

(the $\Rightarrow$ direction is fairly easy, and I did it, but I'm stuck at the other direction.)

share|improve this question
4  
Actually, it is $\implies$ direction that requires some work, certainly more than the other direction. –  André Nicolas Oct 30 '12 at 19:09
add comment

2 Answers

up vote 4 down vote accepted

$x$ and $y$ both satisfy the equation $X^2-(x+y)X+xy$, so they are algebraic over $F(xy, x+y)$. Now recall that if $F\subset K \subset L $ and $L/K$ and $K/F$ are algebraic, so is $L/K$. (You can prove this by using an argument about dimension, since every algebraic element is contained in a finite extension.)

share|improve this answer
    
You meant $X^2 -(x+y)X+xy$ –  Lubin Oct 30 '12 at 19:37
    
Indeed I did. Just fixed it. –  Brett Frankel Oct 30 '12 at 20:54
add comment

Look at the roots of $$\alpha^2 - (x+y) \alpha + xy$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.