> For 12 b, can we show that x^3+ax+b has a repeated root instead, > because that's what we need in c to show that f has nonzero > discriminant. Yes, but remark that it is well-known that a cubic has a repeated root if and only if it has zero discriminant.
On Saturday 11 January 2003 06:11 pm, you wrote: > perfect square. The problem is that MAGMA doesn't have the precision > to compute ceiling(sqrt(n)). Is there a way to do this? Check out http://magma.maths.usyd.edu.au/magma/htmlhelp/text565.htm#4679
> In 5a, a and b are to be integers, right? Yes.
> In 5b, does the sum of the divisors include 1? On the P.S. with a > similar question, that function included 1 and n, but in this case > obviously doesn't include n. sigma is the sum of the proper divisors, so sigma(1)=1, sigma(2)=1, sigma(3)=2, sigma(6)=1+2+3.
> For this, you want some rational number a/b, a,b integers, such that > a/b expanded in decimal is equal to the given floating point up to the > accuracy given, not absolutely equal, right? Answer the problem as stated using the rules at the beginning of the exam. If you think it is impossible to find such a rational number, prove it to get full credit.
> I'm working on problem 5, part b. It's clear that I can use the > algorithm that I devised in the homework assignment a few months ago in > which we have sigma(n) and phi(n) and we want to factor it into pqr. > But I've done everything in the exact same way that I did before, using > MAGMA, and when I factor the polynomial to get the factorization, I get > 2, some really big negative number with a nontrivial decimal, and 1. > This is really weird. I'm just trying to check and see that there's not > a typo in the number or anything? I'm not going to tell you whether or not there is a typo. However, if you can carefully prove that there must be a typo in the number then you will get full credit for the problem.
> One more thing...if we do more than 8 problems, will you take the best > 8 solutions, or will you just ignore the additional ones (i.e. should I > try to do a couple extra if I have time)? Turn in solutions to exactly eight problems. If you turn in more than eight, I will grade the first eight lowest numbered problems. E.g., if you turn in 1-7, 9, 10, and 12, I will discard 10 and 12.
> For problem 3 ("characterize the numbers n such that Z/n is a field", > etc.) do we have to prove our answers? I assume so, but just checking. Yes, but for this problem you can be very terse and cite results from the course notes.
> I'm trying to factor these numbers in number 5, and I see you've > changed the ECM intrinsic that you used in class to perform ECM on > large numbers. I tried "ECM;" but I don't understand what step ratio > and number of steps arguments should be in order for the function to be > effective (if it even is effective?) on the number in 5a. Can you > explain what these arguments are? No comment... except that we discussed many factoring methods besides ECM.
> on p. 12.2 on the final, should it be 4a^3+27b^2=0 or 4a^3-27b^2=0? It's technically a typo, but it doesn't make any difference since if 4a^3-27b^2 = 0 then -(4(-a)^3+27b^2)=0 so 4(-a)^3 + 27b^2 = 0. I.e., there is a bijection between the (a,b) for 4a^3+27b^2=0 and for 4a^3-27b^2=0 given by (a,b) <--> (-a,b). > in p. 7, where are x_0, y_0, \alpha_1, etc. supposed to live? I'll let you decide; make the choice that is most interesting to a mathematician. For example, in 7(a) if the alpha_i, beta_i are only required to lie in the complex numbers then the statement is trivial, hence not interesting.