Re: commutative algebra

In article <0ed85346-b6ae-44e7-8ce5-710ecf932c24@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<sanchopancho80@xxxxxx> wrote:
On 4 Mai, 19:36, Timothy Murphy <gayle...@xxxxxxxxxx> wrote:
(1 + sqrt{-5}, 2) ?

Thank you for your answer. Is it obvious that this is a prime ideal?

Well, depends ->where<-... But presumably Timothy is talking about
Z[sqrt(-5)].

Yes, it is not hard to check that it is a prime ideal (in fact, a
maximal ideal). Of course, if you know something about quadratic
number fields, then this is just the unique prime ideal P lying over
(2). Or you could check that the quotient of Z[sqrt(-5)] by this ideal is
isomorphic to the field of 2 elements.

Or...

Let (a+b*sqrt(-5))(c+d*sqrt(-5)) be an element of P=(1+sqrt(-5),2).

(a+b*sqrt(-5))(c+d*(sqrt-5)) = (ac+5bd) + (ad+bc)*sqrt(-5).

Note that 2*(ad+bc)sqrt(-5) is also in the ideal, hence so is

(ac+5bd)-(ad+bc)*sqrt(-5). Thus,

(ac+5bd)^2 + 5(ad+bc)^2 = (a^2+5b^2)(c^2+5d^2) is in the ideal, and an
integer.

Now, what are the elements of P/\Z? If

2*(r+s*sqrt(-5)) + (u+v*sqrt(-5)(1+sqrt(-5))

is in P/\Z, then you have

(2r+2s*sqrt(-5)) + ((u+5v) + (u+v)*sqrt(-5))
= (2r + u+5v) + (2s+u+v)*(sqrt(-5)) in Z.

Thus, 2s+u+v=0, so u+v is even. Thus, (2r+u+5v) = 2r + 4v + (u+v) =
2r+4v-2s is also even. Hence, P/\Z is contained in 2Z. Conversely, it
is plain that 2Z is contained in P, so P/\Z=2Z.

Thus, from the fact that (a+b*sqrt(-5))(c+d*(sqrt-5)) is in the ideal,
we deduce that (a^2+5b^2)(c^2+5d^2) is even, and therefore that either
(a^2+5b^2) is even or (c^2+5d^2) is even. Without loss of generality,
assume that a^2 + 5b^2 is even. Then either a and b are both even (in
which case (a+b*sqrt(-5)) is in P), or else a and b are both odd. We
can then write b = a + 2k for some integer k, and we have

a+b*(sqrt(-5)) = [(a+a*sqrt(-5))] + 2k*sqrt(-5)
= a(1+sqrt(-5)) * k*sqrt(-5)(2),

which is clearly in P. Thus, P is a prime ideal of Z[sqrt(-5)].

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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