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Revision difference : integral basis of quadratic field |
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Version 2 |
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| Let $m$ be a squarefree integer $\neq 1$. All numbers of the quadratic field $\mathbb{Q}(\sqrt{m})$ may be written in the form |
Let $m$ be a squarefree integer $\neq 1$. All numbers of the quadratic field $\mathbb{Q}(\sqrt{m})$ may be written in the form |
| \begin{align} |
\begin{align} |
| \alpha = \frac{j+k\sqrt{m}}{l}, |
\alpha = \frac{j+k\sqrt{m}}{l}, |
| \end{align} |
\end{align} |
| where $j,\,k,\,l$ are integers with\, $\gcd(j,\,k,\,l) = 1$\, and\, $l > 0$.\, Then $\alpha$ (and its algebraic conjugate \,$\alpha' = \frac{j-k\sqrt{m}}{l}$) satisfy the equation |
where $j,\,k,\,l$ are integers with\, $\gcd(j,\,k,\,l) = 1$\, and\, $l > 0$.\, Then $\alpha$ (and its algebraic conjugate \,$\alpha' = \frac{j-k\sqrt{m}}{l}$) satisfy the equation |
| \begin{align} |
\begin{align} |
| x^2+px+q = 0, |
x^2+px+q = 0, |
| \end{align} |
\end{align} |
| where |
where |
| \begin{align} |
\begin{align} |
| p = -\frac{2j}{l}, \quad q = \frac{j^2-k^2m}{l^2}. |
p = -\frac{2j}{l}, \quad q = \frac{j^2-k^2m}{l^2}. |
| \end{align} |
\end{align} |
| We will find out when the number (1) is an algebraic integer, i.e. when the coefficients $p$ and $q$ are rational integers. |
We will find out when the number (1) is an algebraic integer, i.e. when the coefficients $p$ and $q$ are rational integers. |
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| Naturally, $p$ and $q$ are integers always when\, $l = 1$.\, We suppose now that\, $l > 1$.\, The latter of the equations (3) says that $q$ can be integer only when |
Naturally, $p$ and $q$ are integers always when\, $l = 1$.\, We suppose now that\, $l > 1$.\, The latter of the equations (3) says that $q$ can be integer only when |
| $$(\gcd(j,\,l))^2 = \gcd(j^2,\,l^2) \mid k^2m.$$ |
$$(\gcd(j,\,l))^2 = \gcd(j^2,\,l^2) \mid k^2m.$$ |
| Because\, $\gcd(j,\,k,\,l) = 1$,\, we have by the corollary of B\'ezout's lemma that\, $\gcd(j,\,l) \mid m$.\, Since $m$ is squarefree, we infer that |
Because\, $\gcd(j,\,k,\,l) = 1$,\, we have by the corollary of B\'ezout's lemma that\, $\gcd(j,\,l) \mid m$.\, Since $m$ is squarefree, we infer that |
| \begin{align} |
\begin{align} |
| \gcd(j,\,l) = 1. |
\gcd(j,\,l) = 1. |
| \end{align} |
\end{align} |
| In order that also $p$ were an integer, the former of the equations (3) implies that\, $l = 2$. |
In order to also $p$ were an integer, the former of the equations (3) implies that\, $l = 2$. |
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| So, by the latter of the equations (3),\, $4 \mid j^2-k^2m$, i.e. |
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| \begin{align} |
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| k^2m \equiv j^2 \pmod{4}. |
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| \end{align} |
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| Since by (4),\, $\gcd(j,\,2) = 1$,\, the integer $j$ has to be odd.\, In order that (5) would be valid, also $k$ must be odd.\, Therefore,\, $j^2 \equiv 1 \pmod{4}$\, and\, $k^2 \equiv 1 \pmod{4}$,\, and thus (5) changes to |
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| \begin{align} |
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| m \equiv 1 \pmod{4}. |
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| \end{align} |
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| If we conversely assume (6) and that $j,\,k$ are odd and\, $l = 2$, then (5) is true, $p,\,q$ are integers and accordingly (1) is an algebraic integer. |
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| We have now obtained the following result: |
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| \begin{itemize} |
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| \item When\, $m \not\equiv 1 \pmod{4}$,\, the integers of the field $\mathbb{Q}(\sqrt{m})$ are |
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| $$a+b\sqrt{m}$$ |
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| where $a,\,b$ are arbitrary rational integers; |
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| \item when\, $m \equiv 1 \pmod{4}$,\, in \PMlinkescapetext{addition} to the numbers $a+b\sqrt{m}$, also the numbers |
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| $$\frac{j+k\sqrt{m}}{2},$$ |
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| with $j,\,k$ arbitrary odd integers, are integers of the field. |
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| \end{itemize} |
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| Then, it may be easily inferred the |
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| \textbf{Theorem.}\, If we denote |
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| \begin{align*} |
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| \omega := |
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| \begin{cases} |
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| & \frac{1+\sqrt{m}}{2} \quad \mbox{when } m \equiv 1\pmod{4},\\ |
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| & \sqrt{m} \quad \mbox{ when } m \not\equiv 1\pmod{4}, |
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| \end{cases} |
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| \end{align*} |
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| then any integer of the quadratic field $\mathbb{Q}(\sqrt{m})$ may be expressed in the form |
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| $$a+b\omega,$$ |
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| where $a$ and $b$ are uniquely determined rational integers.\, Conversely, every number of this form is an integer of the field.\, One says that 1 and $\omega$ form an integral basis of the field. |
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| \begin{thebibliography}{9} |
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| \bibitem{K.V.} {\sc K. V\"ais\"al\"a}: {\em Lukuteorian ja korkeamman algebran alkeet}.\, Tiedekirjasto No. 17.\quad Kustannusosakeyhti\"o Otava, Helsinki (1950). |
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| \end{thebibliography} |
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