Notation of $(2, \frac{1 + \sqrt{d}}{2})$ and its Norm

Asked Feb 02 '22 at 22:27

Active Feb 02 '22 at 22:46

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I am lacking some really basic algebra knowledge.

Can somebody help me with the following notation in $\mathbb{Q}[\sqrt{d}]$?

$$ (2)=(2, \frac{1 + \sqrt{d}}{2})(2, \frac{1 - \sqrt{d}}{2}) $$

Does it mean $$(2, \frac{1 + \sqrt{d}}{2})(2, \frac{1 - \sqrt{d}}{2}) = (2 + \frac{1 + \sqrt{d}}{2})(2 + \frac{1 - \sqrt{d}}{2}) \\ = 4 + (\frac{1 + \sqrt{d}}{2})( \frac{1 - \sqrt{d}}{2}) ~ ??$$

And how do I calculate the Norm of $(2, \frac{1 + \sqrt{d}}{2})$?

$$ N\left((2, \frac{1 + \sqrt{d}}{2})\right) = N\left((2 + \frac{1 + \sqrt{d}}{2})\right) = ? $$

edited Feb 02 '22 at 22:46

asked Feb 02 '22 at 22:27

Alex

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The ideal generated by those elements and the norm of this ideal? Does this make sense in the context of your problem? – includeCMath Feb 02 '22 at 22:30
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$(2, \frac{1 + \sqrt{d}}{2})$ probably denotes the ideal generated by $2$ and $\frac{1+\sqrt{d}}{2}$. – jjagmath Feb 02 '22 at 22:32
It is from here (the last comment) https://math.stackexchange.com/questions/382188/showing-that-mathbb-q-sqrt17-has-class-number-1. I am following an algebraic number theory course and I do not know lots about algebra, so I am struggling with the notations and I do not find it explained, since it is so trivial – Alex Feb 02 '22 at 22:34
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Then that's what @jjagmath and I suggested – includeCMath Feb 02 '22 at 22:37
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I upvoted the question because of the first line "I am lacking some really basic algebra knowledge". – markvs Feb 02 '22 at 22:56
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That the ideal product $,(2,w)(2,\bar w) = (2)$ follows from an ideal form of the Four Number Theorem, namely $,d\equiv 1\pmod{!8}\Rightarrow \color{#c00}{2\mid w\bar w}=(1!-!d)/4,$ thus $$(2,w)(2,\bar w) = (2)(2,\color{#0a0}{w,\bar w},\color{#c00}{w\bar w/2})= (2)\ \ {\rm by}\ \ ,\color{#0a0}{w+\bar w} = 1\qquad$$ – Bill Dubuque Feb 03 '22 at 08:04
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See also this answer – Bill Dubuque Feb 03 '22 at 08:23
Thank you very much! That is very helpful! – Alex Feb 03 '22 at 10:00

Notation of $(2, \frac{1 + \sqrt{d}}{2})$ and its Norm

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