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For any positive integer $k$ and prime $p$ find necessary and sufficient condition for $Z_p[\sqrt k] =\{a+b{\sqrt k}\mid a,b \in Z_p\}$ to be a field. Any help would be appreciated. While I am getting for any prime $p$ and $k.$ I don't think it's right. Thanks.

Michael Hardy
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a math lover
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  • You're correct: your conjecture is wrong. For example, with $p=2$ and $k=3$ it is not a field. Maybe if you said something about how you got to your guess, we can help. – rschwieb Jan 30 '18 at 16:23
  • I know. This was the example – a math lover Jan 30 '18 at 16:59
  • How are operations defined in $\mathbb Z_p[\sqrt k]$ ? What does $\sqrt k$ mean in the context of $\mathbb Z_p$ ? – lhf Jan 30 '18 at 21:54

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For me, the correct interpretation of $\mathbb Z_p[\sqrt k]$ is $ \mathbb Z_p[X]/(X^2-k) $.

With this interpretation, we have $\mathbb Z_p[\sqrt k]$ is a field iff $k$ is not a square mod $p$, because when $k \equiv a^2 \bmod p$ we have $$ \mathbb Z_p[X]/(X^2-k) = \mathbb Z_p[X]/(X^2-a^2) \cong \mathbb Z_p[X]/(X-a) \times \mathbb Z_p[X]/(X+a) \cong \mathbb Z_p \times \mathbb Z_p $$

lhf
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