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November 16th, 2006, 07:12 AM   #1
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GCD question

Find the GCD of 24 and 49 in the integers of Q[sqrt(3)], assuming that the GCD is defined. (Note: you need not decompose 24 or 49 into primes in Q[sqrt(3)].

Please help me!
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November 16th, 2006, 07:35 AM   #2
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Hint? If u divides x and y then u divides mx+ny for all ordinary integers m and n.
Larry Hammick is offline  
November 16th, 2006, 10:29 AM   #3
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Hi Larry,

Could you please show me how? I got stuck with this question for long. It never state in my book about doing the GCD of a quadratic integer, but it turns out that I have to do this question.

Honestly, I don't really get it even with your hint. Please kindly show me the way. Thank you very much.
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November 16th, 2006, 11:47 AM   #4
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GCD

A common divisor of 24 and 49
will also divide 49 - 24
and 49 - 2*24
Einar is offline  
November 16th, 2006, 12:27 PM   #5
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The way you showed me is by Euclidean Algorithm. The gcd end up to be 1.

But the question is asking to find the GCD OF 24 AND 49 IN THE INTEGERS OF Q[sqrt(3)]. Not only to find the GCD of 24 and 49.....
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November 16th, 2006, 09:59 PM   #6
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OK, let d be a common divisor of 24 and 49
Then, for some e,f
24 = de, 49 = df
Then we get
df - 2de = 1
giving that
d(f-2e) = 1
Thus a common divisor of 24 and 49 must be a divisor of 1
A divisor of 1 is called a unit, and when we search for a GCD, two common
divisors are considered equal if their quotient is a unit. So the GCD is 1.
Two things should be noted here:
1) Although there might not be a euclidean algorithm in a ring, it is always
allowed to try, and if the algorithm ends after finitely many steps, you may use the result.
2) There ARE non-trivial units in Z(sqrt(3)), all the numbers
+- (2+sqrt(3))^n, n any (positive or negative) integer.
Einar
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