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So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things:

1) There exists an element a of $\mathbb{F}$ that's not in $\mathbb{Q}$ such that it satisfies the equation $a^2-n=0$ for some $n\in\mathbb{Z}$.

2) That $\mathbb{F}$ is isomorphic to $\mathbb{Q}[\surd(n)]$ and further that $n$ is square-free.

So, showing there's an element in the complement of $\mathbb{F}$ and $\mathbb{Q}$ I've managed, but I can't show that it satisfies the equation in question. For number 2, I'm lost.

(Trying LaTeX, hope I didn't screw it up too bad.)

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    There are many elements in $\mathbb F \setminus \mathbb Q$. Not all of them satisfy $a^2-n=0$. The point is that given some $b \in \mathbb F \setminus \mathbb Q$ you can find$a$related $a$ that works.2012-09-21

1 Answers 1

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A start: Let $\alpha\in \mathbb{F}$, with $\alpha\not\in \mathbb{Q}$. Argue that by dimensionality considerations, $\alpha^2$, $\alpha$, and $1$ are linearly dependent over the rationals.

This shows that $\alpha$ is the root of a quadratic with rational (or equivalently integer) coefficients.

Complete the square, or use the Quadratic Formula, to produce the requisite $n$ (think discriminant).

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    Ah, okay. Thanks you original combination of original members of the Bourbaki troupe!2012-09-21