Re: Algebra with gaussian integer.

Mariano Suarez-Alvarez <mariano.suarezalvarez@xxxxxxxxx> wrote:
On Aug 13, 1:33?am, "mina_world" <mina_world@xxxxxxxxxxx> wrote:

Z[i] / <2 + 2i>

Find the elements and order.
---------------------------------------------------------
(1 - i)(2 + 2i) = 2 + 2i - 2i +2 = 4 ?in <2 + 2i>

[lots of computation omitted]

so, Z[i] / <2 + 2i> = {0, 1, 2, 3, i, 1 + i, 2 + i, 3 + i}
-------------------------------------------------------------
OR I can use grid.

[lots of computation omitted]

so, Z[i] / <2 + 2i> = {0, i, 2i, 3i, 1 + i, -1 + i, 1 + 2i, -1 + 2i}

The order of Z[i] / (a + b i) is simply a^2 + b^2.
To show that, observe that you are trying to compute
the order of the cokernel of the map

Z[i] --> Z[i]

given by multiplication by a + b i. In the Z-basis {1, i}
of Z[i], the matrix of this map is

(a -b)
(b a)

In general, the cokernel of an injective map Z^2 --> Z^2
with matrix

(a b)
(c d)

has order |ad-bc|.

Or one may use the standard criterion for a module to be an ideal,
e.g see Prop. 2.8 in [1]. This implies that (4,2+2i) = [4,2+2i]
and the latter module has obvious norm 4*2, see Prop. 2.9.
This criterion extends to general Hermite normal form matrices,
see section 4.7.2 of Henri Cohen's: A Course in Computational
Number Theory.

--Bill Dubuque

[1] Franz Lemmermeyer. Ideals in quadratic number fields.
http://www.fen.bilkent.edu.tr/~franz/ant/ant02.pdf
.

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