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)].

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 teach me . Thank you very much.

The only integer divisor of both 24 and 49 is 1. I don't know what you mean by Q[sqrt(3)], nor the "integers of Q"

I can see why you are confused by the question. So am I. Are you sure you are stating it accurately? Perhaps if you explain what Q[sqrt(3)] is, someone can help you

I'm not sure if Euclid's algorithm is still valid for algebraic integers. If it is, then the GCD should be 1. If you use the rule:

GCD[a,b] = GCD[a, b Mod a],

you fund:

GCD[24,49] = GCD[24, 49 - 2*24] =
GCD[24,1] = 1

um... we notices that if just finding the GCD of 24 and 49, the answer is 1.

But here is about GCD of them in the integers of Q[sqrt(3)]. Are they the same?

Q mean quadratic integers. Quadratic integers are in the form of a+bsqrt(d). In here d=3.

Yes, I think it is different for algebraic numbers. I never studied this subject properly...

me too. I am taking the number theory , but the ending chapter is about quadratic integer. The lecture and book do not go in deep for this topic , but the questions are. :(

Answer 1

I understand your frustration. Finding the GCD of numbers in the integers of Q[sqrt(3)] can be a bit more complicated than finding the GCD of regular integers. In this case, we need to consider numbers of the form a + b√3, where a and b are integers.

To find the GCD of 24 and 49 in the integers of Q[sqrt(3)], we can use the Euclidean algorithm. The Euclidean algorithm states that if we have two numbers, a and b, the GCD of a and b is equal to the GCD of a and (b mod a).

Let's apply the Euclidean algorithm to find the GCD of 24 and 49 in the integers of Q[sqrt(3)]:

GCD(24, 49) = GCD(24, 49 - 2√3 * 24)
= GCD(24, 49 - 48√3)
= GCD(24, 49 - 48√3)
= GCD(24, 49 - 48√3)
= GCD(24, 1)

Now, let's find the GCD of 24 and 1 in the integers of Q[sqrt(3)]:

GCD(24, 1) = GCD(24 - 1 * 24, 1)
= GCD(24 - 24, 1)
= GCD(0, 1)

Since the second number is nonzero, we can conclude that the GCD of 24 and 49 in the integers of Q[sqrt(3)] is 1.

Please note that I am not able to provide a step-by-step explanation for calculating GCD in the integers of Q[sqrt(3)], as it is more complex and requires deeper knowledge of quadratic integers. I recommend consulting a textbook or academic resources specifically focused on algebraic number theory for a more comprehensive explanation.