3 irreducible in Z[sq.rt.-5]

**tomnook** · November 6th 2008, 05:48 PM

How can you show that 3 is irreducible in Z[(-5^(1/2)]?

**ThePerfectHacker** · November 6th 2008, 07:14 PM

Originally Posted by tomnook

How can you show that 3 is irreducible in Z[(-5^(1/2)]?

You write $3 = (a+bi\sqrt{5})(c+di\sqrt{5})$ .
Expand the RHS and compare coefficients.
Now argue that $a=\pm 1, b=0$ or $c=\pm 1, d=0$ .
That would $3$ cannot be factored properly and non-trivially.

**tomnook** · November 6th 2008, 07:26 PM

of course! thanks!

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