A Prime Ideal in the Ring $\Z[\sqrt{10}]$

Problem 223

Consider the ring
$\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}$ and its ideal
$P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.$ Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.

Definition of a prime ideal.

An ideal $P$ of a ring $R$ is a prime ideal if whenever we have $ab \in P$ for elements $a, b \in R$, then either $a\in P$ or $b \in P$.

Proof.

Suppose that $a+b\sqrt{10}, c+d\sqrt{10} \in \Z[\sqrt{10}]$ and the product
$(a+b\sqrt{10}) (c+d\sqrt{10}) \in P.$ Then expanding the product, we have
$ac+10bd+(ad+bc)\sqrt{10} \in P.$

Since $ac+10bd$ must be even number, we have either $a$ or $c$ is even.
Hence either
$a+b\sqrt{10}\in P \text{ or } c+d\sqrt{10} \in P,$ and we conclude that $P$ is a prime ideal.

Further Question.

In fact, it can be proved that $P$ is a maximal ideal in the ring $\Z[\sqrt{10}]$.

For a proof, see the post “Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$“.

1 Response

1. 06/27/2017

[…] A direct proof that the ideal $P=(2, sqrt{10})$ is prime in the ring $Z[sqrt{10}]$ is given in the post “A prime ideal in the ring $Z[sqrt{10}]$“. […]

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