- #1
ElDavidas
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Homework Statement
Take the ideal
[tex] I = < 6, 3 + 3 \sqrt{-17} >[/tex]
in the ring [itex]Z [ \sqrt{-17} ][/itex].
Determine whether this ideal is prime or not.
Homework Equations
[tex]<18> = I^2 [/tex]
There is no element [itex]\alpha \in Z [ \sqrt{-17} ] [/itex] such that [itex] 18 = \alpha^2[/itex]
The Attempt at a Solution
I really don't know how to go about doing this. I have the definition of a prime ideal P.
[itex] P [/itex] is a prime ideal [itex] \Leftrightarrow [/itex] if [itex] ab \in P[/itex] then [itex] a \in P [/itex] or [itex] b \in P [/itex].
And I see that [itex] < 6, 3 + 3 \sqrt{-17} > = 6 Z [ \sqrt{-17} ] +( 3 + 3 \sqrt{-17}) Z [ \sqrt{-17} ][/itex].
Is it also possible to pull out the [itex] Z [ \sqrt{-17} ][/itex] from the above equation?
Thanks
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