- #1
MissMoneypenny
- 17
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Homework Statement
Let S={p+qα+rα2 : p, q, r [itex]\in \mathbb{Q}[/itex]}, where α=[itex]\sqrt[3]{2}[/itex]. Then S is a subfield of [itex]\mathbb{C}[/itex]. Prove that each nonzero element of S has a multiplicative inverse in S.
The Attempt at a Solution
Let p, q, r[itex]\in\mathbb{Q}[/itex] 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 [itex]\mathbb{Q}[/itex]. 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!