None of this nursery school stuff - a proper maths problem. 25$ reward

[PairTheBoard]

More problems like this please:

http://forumserver.twoplustwo.com/sh...mp;sb=5&o=

PairTheBoard

[pzhon]

[ QUOTE ]

Lets let A = x + 2.sqrt(-17).

Now it is possible to show, by considering the factorisation of y^5, that A itself is a fifth power in this field. This is difficult.

[/ QUOTE ]
Is there unique prime factorization in Z[sqrt(-17)]?

My approach was to note that y must be divisible by 11, then to try to factor the ideal <11>, with the idea of trying to show that A is divisible by 11. However, I believe the ideal <11> factors by duality, since x^2+17 factors mod 11.

[jason_t]

[ QUOTE ]
x^2 + 68 = y^5 then

x^4 + 136x^2 + 68^2 = y^10

Now working modulo 11

By Little Fermat theorem, y^10 = 1 if (y,11)=1

68=2 68^2=4 136 = 4 then

x^4 + 4x^2 + 4 = 1 mod 11

(x^2 + 2)^2 = 1 mod 11

therefore x^2 + 2 = 1 so x^2 = -1 mod 11

or x^2 + 2 = -1 this is x^2 = -3 = 8 mod 11

contradiction in both cases!!!

Now, if y = 11k then y = 1 mod 10

so, x^2 + 68 = 1 mod 10

x^2 - 2 = 1 mod 10

x^2 = 3 mod 10 contradiction !!!

Q.E.D.

David

[/ QUOTE ]

First, I think this the right approach and what I was trying earlier but

y = 11k then y = 1 mod 10

is incorrect

and I think we should be working modulo 17 (prime factor of 68), not modulo 11.

[jason1990]

[ QUOTE ]
Is there unique prime factorization in Z[sqrt(-17)]?

[/ QUOTE ]
Theorem 8.22 of An Introduction to Number Theory, Harold M. Stark:
If d<0, then Q(\sqrt{d}) has the unique factorization property if and only if d is one of the nine numbers -1,-2,-3,-7,-11,-19,-43,-67, and -163.

[Rotating Rabbit]

[ QUOTE ]
[ QUOTE ]
Is there unique prime factorization in Z[sqrt(-17)]?

[/ QUOTE ]
Theorem 8.22 of An Introduction to Number Theory, Harold M. Stark:
If d<0, then Q(\sqrt{d}) has the unique factorization property if and only if d is one of the nine numbers -1,-2,-3,-7,-11,-19,-43,-67, and -163.

[/ QUOTE ]

Exactly. Unfortunately, -17 not equal to 1 mod 4, which effectively destroys any chance of the ring of integers of Q(sqrt(-17)) being a UFD.

(Briefly, this is because the ring is too small. Note that if a = c + d.sqrt(-17) and b = c - d.sqrt(-17) then (X - a)(X - b) = X^2 - 2cX + c^2 - 17d^2 as the minimum polynomial doesnt have many roots as T =( 1 + sqrt(-17) / 2 ) doesnt solve the above). So finding a euclidean norm is impossible here. For positive values in the sqrt matters get even worse...)

But its still possible to navigate around this. But as I said this problem is way too hard, which is my fault.