Expert: Paul Klarreich - 11/23/2008

Question
The problem statement: If a in K is such that a^2 is algebraic over the subfield F of K, show a is algebraic over F.

definition of algebraic: a is algebraic over a field F is there exists p(a) = 0 for p(x) in F[x].

I know if a^2 is algebraic over F then x^2-a^2= 0 which factors as (x-a)(x+a) = 0 so a is a zero of x^2 -a^2. My issue is I don't know how to explain how all this ties into the answer. Any suggestions?

Sombra

Answer
Questioner:   Sombra
Category:  Advanced Math
Private:  No
 
Subject:  fields
Question:  The problem statement: If a in K is such that a^2 is algebraic over the subfield F of K, show a is algebraic over F.

definition of algebraic: a is algebraic over a field F if there exists p(a) = 0 for p(x) in F[x].

I know if a^2 is algebraic over F then x^2-a^2= 0 which factors as (x-a)(x+a) = 0 so a is a zero of x^2 -a^2. My issue is I don't know how to explain how all this ties into the answer. Any suggestions?
................................
Hi, Sombra,

You know that there exists a polynomial for which a^2 is a zero:

P(a^2) = 0  

Let's say P(x) is quadratic.  [I just don't feel like typing any more than that.]

P(x) = bx^2 + cx + d

P(a^2) = b(a^2)^2 + c(a^2) + d = 0

which implies:

ba^2(a^2) + ca(a) + d = 0

But I think that  ba^2, ca, d, are all elements of your subfield.  

Write  Q(x) = ba^2(x^2) + ca(x) + d

This is a polynomial over K for which x = a is a zero.

Does that help?