Let $\; p,d \,$ be prime numbers I wanna proof that $\; M:=\{a_o+a_1\sqrt p +a_2\sqrt d +a_3\sqrt {pd}:a_o,a_1,a_2,a_3 \in \mathbb Q \}\;$ is a field. The only thing I am having trouble with is finding the inverse of $p \in M$. Is it true that $\;M\;$ is a field and if so how would one find the inverse of an element. Give me a little hint.
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A special case of this question. – Jyrki Lahtonen Dec 20 '15 at 16:40
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Hint: prove that $\exists w,x,y,z$ so that $(a_0+a_1\sqrt{p}+a_2\sqrt{q}+a_3\sqrt{pq})*(w+x\sqrt{p}+y\sqrt{q}+z\sqrt{pq}) = 1$. There will be a system of four linear equations, you must prove it has only one solution. – Abstraction Dec 20 '15 at 16:40
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@Abstraction: While that works in principle, you then face the difficulty of proving that the determinant of the resulting system is non-zero. – Jyrki Lahtonen Dec 20 '15 at 16:42
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You could try rationalizing the fraction $\frac{1}{a_0 + a_1 \sqrt{p} + \cdots}$. Write it as $\frac{1}{b + c \sqrt{d}}$ where $b$, $c$ are expressions involving $\sqrt{p}$. Multiply both numerator and denominator by ${b - c \sqrt{d}}$ to get rid of $\sqrt{d}$ in the denominator. Then for the new equivalent fraction fraction, do a similar thing to get rid of $\sqrt{p}$ in the denominator. – orangeskid Dec 20 '15 at 17:27
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In the post Proof "that an integral domain that is a finite-dimensional F-vector space is in fact a field " Georges Elencwajg gives a short proof with a map $\phi :M\rightarrow M ; x\rightarrow rx;$. $\phi;$ is injective which is obvious, but why is it surjective ? – XPenguen Dec 20 '15 at 18:45