A GCD question- urgent help!
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.
Suppose 32 = alpha*beta for alpha, beta reatively prime quadratic integers in Q[i] . Show that alpha = epsilon*gamma^2 for some unit epsilon and some quadratic integers gamma in Q[i].
If alpha is a quadratic integers in Q[sqrt(-d)] , then define a notion of congruence (mod alpha).
Furthermore, define +, -, and X for congruence classes , and show that this notion is well-defined.
Could you please guide me to begin this question? Thank you very much.
Find the prime factorization of 6 in Q[sqrt(-1)].
Ans : Since 6 = 2*3
so 6 = (1+sqrt(-1)) (1-sqrt(-1)) *3 Q.E.D.
Do I need to add anything to it? Am I done with this question? Please kindly advise me. Thank you very much.
Consider Q[sqrt(2)].
Does every element of Q[sqrt(2)] have a square root in Q[sqrt(2)] ?
Prove if true, and give a counterexample if false.My solution:
sqrt(sqrt(2)) = a + bsqrt(2)
if I square both sides then I will have :
sqrt(2) = (a + b*sqrt(2))^2
= a^2 + 2ab*sqrt(2) + 2b^2...
Can anyone guide me how to prove this question??
Question:
We want to describe via a picture a set of subsets of a square which are something like diagonals, but are not quite the same. We'll call them steep diagonals. One of them, labelled e, is illustrated in the square below; the other 6...
No , don't know what to go on next?
Does the below theorem useful for go on my prove?
if p is prime, the equation x^2 ≡ - 1 mod p has solution iff p ≡ 1 mod 4