Proving each nonzero element of a subfield of C has an inverse

1. Sep 12, 2014

MissMoneypenny

1. The problem statement, all variables and given/known data

Let S={p+qα+rα2 : p, q, r $\in \mathbb{Q}$}, where α=$\sqrt[3]{2}$. Then S is a subfield of $\mathbb{C}$. Prove that each nonzero element of S has a multiplicative inverse in S.

3. The attempt at a solution

Let p, q, r$\in\mathbb{Q}$ such that not all of p, q, r are zero. If each nonzero element of S has a multiplicative inverse in S, then there are rational numbers a, b, c so that

(p+qα+rα2)(a+bα+cα2)=1

Expanding and using the fact that α3=2 and α4=2α, we get:

(pa+2qc+2rb)+α(pb+qa+2rc)+α2(pc+qb+ra)=1

Since 1, α, and α2 are linearly independent we may equate coefficients to give a system of equations:

pa+2qc+2rb=1
pb+qa+2rc=0
pc+qb+ra=0

Now to finish the problem I want to show that this system always has a solution in $\mathbb{Q}$. I tried using Cramer's rule (i.e. showing the determinant of the matrix whose entries are the coefficients of p, q, r is equal to zero only if p=q=r=0), but I had no luck and it just turned into a mess. Likewise, back substitution was a mess. Can anyone give me a hand on how to complete this problem?

Thanks!

2. Sep 12, 2014

Ray Vickson

If $x, y \in S$ have multiplicative inverses in $S$ and if $x + y \neq 0$, can you prove that $x+y$ has a multiplicative inverse in $S$? Can you prove that $1, \alpha, \alpha^2$ all have multiplicative inverses in $S$?

3. Sep 12, 2014

MissMoneypenny

Thanks for the quick response! I think I see what you're getting at. If I can prove what you wrote, then since each element of S is a linear combination of 1, α, and α2 whose coefficients are in $\mathbb{Q}$, it follows that each element of S has an inverse in S. Unfortunately I really am not sure how to show that if $x, y, x^{-1}, y^{-1} \in S$ with $x \ne y$, then $(x+y)^{-1} \in S$. I tried to solve the equation $(x+y)(ax^{-1}+bx^{-1})=1$ for a and b, but that didn't get me anywhere, and I'm not sure what else to try. Is it possible for you to give me a small hint to get me going in the right direction?

Thanks again.

4. Sep 13, 2014

HallsofIvy

Staff Emeritus
Was that really the original statement of the problem? Saying "S is a subfield of $mathbb{C}$. Prove that each nonzero element of S has a multiplicative inverse in S" seems very strange! A subfield is, by definition, a field so each non-zero element must have a multiplicative inverse. It would make more sense to first define S then ask to show that every nonzero element has a multiplicative inverse, as part of proving that S is a subfield.

5. Sep 13, 2014

MissMoneypenny

You're correct, what I wrote is a strange problem statement. It was my fault. The full problem was to show S is a subfield of C. However, I knew how to prove all of the other field axioms hold for S, so I changed the problem statement. As you correctly pointed out, I didn't change it very well. Would it be better if I wrote "confirm that each nonzero element of S has an identity"? Or should I just have written up the actual problem statement but mentioned I only need help proving the existence of multiplicative inverses?

6. Sep 13, 2014

Ray Vickson

Many of these issues are dealt with in
http://math.stackexchange.com/quest...proof-that-the-algebraic-numbers-form-a-field

If you Google 'algebraic numbers' or 'algebraic number fields' you will encounter loads of material on your topic.