I am reading Dummit and Foote Chapter 7.(adsbygoogle = window.adsbygoogle || []).push({});

D&F use a quadratic field as an example of a ring. I am trying to get a good understanding of this ring.

D&F define a quadratic field as follows:

Let D be a rational number that is not a perfect square in and define

[tex] \mathbb{Q} ( \sqrt D ) = \{ \ a + b \sqrt D \ | \ a,b \in \mathbb{Q} \ \}[/tex]

as a subset of [itex] \mathbb{C} [/itex]

In this example D&F write ... "... It is easy to show that the assumption that D is not a square implies that every element of [itex]\mathbb{Q} ( \sqrt D ) [/itex] may be written uniquely in the form [itex] a + b \sqrt D [/itex]."

How do you show this? Further, I am not sure why this assumption is needed?

Is it because we have both positive and negative roots of a square number like 4, but then only consider the principal root [itex] + \sqrt 3 [/itex] of 3? This seems slightly inconsistent!

Also how does the above fit with the idea that D must be not only not a perfect square but squarefree? Is the squarefree condition on D necessary? If so why?

Can someone please clarify this situation for me?

PeterMarshall

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# Basic definition of Quadratic Fields

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