I am reading a syllabus about Discrete Mathematics. One of the problems encouraged in the syllabus to solve is the following.

Define $R= \{ a+ b\sqrt{2} : a,b \in \mathbb Q \}$ Find $f$ so $f$ defines an isomorphism between R and $\mathbb Q[x]/(f)$. Any ideas on how to tackle this problem?

  • $\begingroup$ I would consider the minimal polynomial of $\sqrt{2}$, do you know what this means? $\endgroup$ – DanZimm Jun 16 '14 at 12:23
  • $\begingroup$ I know what minimal polynomials are in the context of finite fields. With minimal polynomial do you mean the polynomial of lowest degree that makes $\sqrt{2}$ zero? Don't really see the connection between minimal polynomials and this problem. $\endgroup$ – tim_a Jun 16 '14 at 12:26

$\renewcommand{\phi}{\varphi}$The idea is to consider the unique ring homomoprphism $$ \phi : \mathbb{Q}[x] \to R $$ which satisfies $\phi(a) = a$ for $a \in \mathbb{Q}$, and $\phi(x) = \sqrt{2}$.

Show this is surjective, and find that the kernel is the principal ideal generated by the minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$.

This assumes a bit of knowledge. It can be reformulated in more elementary terms, though. Please advise in case.


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