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I'm studying about the valuation for Euclidean Domains and quadratic fields $\mathbb{Q}(\sqrt{\theta})=\{ \alpha: \alpha=a+b\sqrt{\theta}, a,b \in \mathbb{Q} \}$ and I'm not sure whether we start with the definition of the (field) norm by $N(\alpha)=\alpha \overline{\alpha}$ or $N(\alpha)=|\alpha \overline{\alpha}|$. So I've looked around and I found two books using different definitions:

  • Gallian, Contemporary Abstract Algebra, uses $N(\alpha)=|\alpha \overline{\alpha}|$;
  • Stark, An Introduction to Number Theory, uses $N(\alpha)=\alpha \overline{\alpha}$.

Surely, only the norm with absolute value can be used for an Euclidean valuation (by definition we only allow to be non-negative integers). But I'm still concerned which one is the original definition for this norm?

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In algebraic number theory, usual norm of an element $\alpha$ of a number field $K$ is given by the product of the Galois conjugates of $\alpha$, i.e. $$ N_{K/\mathbb{Q}}(\alpha) = \prod_{\sigma \in \operatorname{Gal}(K/\mathbb{Q})} \sigma(\alpha).$$

For the quadratic field $\mathbb{Q}(\sqrt{\theta})$, this reduces to Stark's definition of $N(\alpha) = \alpha \overline{\alpha}$.

However, you are also correct that when studying norm-Euclidean domains, we usually take $N(\alpha)$ to mean the absolute value of $N_{K/\mathbb{Q}}(\alpha)$, i.e. $N(\alpha) = |\alpha \overline{\alpha}|$.

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