Fields generated by elements

Question

I am revising for my module exam "Fields". Reading through an example "Show $\mathbb{Q}(2^{1/3})$ is a subfield of $\mathbb{R}$ with $\mathbb{Q}$ and $2^{1/3}$. I understand by the minimality we aim to show its a field, using the field axioms but firstly in the example they express $F = \{a1 + bλ + cλ^2: a, b, c ∈ \mathbb Q\} ⊆ \mathbb Q(λ) ⊆ \mathbb{R}$ where $λ=2^{1/3}$ and show $1,λ, λ^2$ are linearly independent over $\mathbb{Q}$. Why is it necessary to show these terms are linearly independent before proving $\mathbb{Q}(2^{1/3})$ is a field?

Answer 1

There are (at least) two ways of defining $\Bbb Q(\lambda)$. If you define it as the smallest field containing $\Bbb Q$ and $\lambda$, this automatically gives you a field. The second way is to look at all numbers of the form $a+b\lambda+c\lambda$, the coefficients being rational numbers. Here, you need to show that the sum, difference, product, and quotient of two numbers this form again has this form. Neither of these definitions uses the linear independence that you mention.

Answer 2

I am trying to show essentially that F=$ \mathbb Q( \lambda ) $ . I know F $ \subseteq $ $\mathbb Q( \lambda) $. So want to show $\mathbb Q (\lambda) \subseteq $ F. Can do so by showing that F is a field using the subfield criterion. In the subfield axioms, there must exist an inverse in F for all elements which are members of F. Thus we need to show $ 1, \lambda, \lambda^2 $ are linearly independent so we know when finding the inverse that we are not dividing by zero. Is this correct?