Recent content by sidm

so i came up with a proof that..well.. Let L/K be a field extension and we have defined the norm and trace of an element in L, call it a, to be the determinant (resp. trace) of the linear transformation L -> L given by x->ax. Now it's well known that the determinant and trace are the...

for a in Q by basically the same reasoning we must have (if a=c/d) that cd is an odd (positive) power of 7 times a nonzero residue modulo 7. Since d^2=7^{2r}A^2 where d=7^mA then even if A is a nonresidue then it's square won't be and thus we have that a= an odd power of 7 (possibly negative)...

solutions (over Q7) to 7x^2-a (for the time being a is an integer) are in one to one correspondence with those of x^2-7a. Now the solutions of the latter will be in Z7 by Gauss's lemma. It is necessarily true that if a solution exists it will exist modulo 7^k for all k so let this k be...

aha! you are correct indeed: in hensel's lemma the factors modulo p need to be relatively prime! which they are not in the case of x^2-7! ok back to square 1. I will think about this more.

Notice that the roots of x^2-c/d are in 1-1correspondence with those of 7x^2-c/d because root(7) is in Q7 (by hensel's lemma since x^2-7 is reducible modulo 7). So we look at when solutions of the former exist. By gauss's lemmma (let a=c/d relatively prime) then a solution to dx^2-c=0 exists...

I will post my solutions to a few problems I've considered here, I just need feedback from people who can tell me if my ideas are correct because I'm feeling shaky about the methods. Z7 is the dvr in the completion of the rationals w.r.t. the 7adic metric. Then for what integers a is there a...

ahhh, this is a good fact to know, much appreciated!

well i have an answer tomy last question: h_n-h_m=p^{n+1}(something) thus the difference between coefficients of say x_i is a multiple of p^{n+1} is thus small. Thus h and g exist and clearly their product is equal to f: f-gh has arbitrarily small absolute value (triangle inequality with...

We're are looking at a field, K, complete with respect to a (normalized nonarchimedean) valuation, ||, and let A be it's discrete valuation ring (all elements of K with absolute value less than or equal to 1) with maximal ideal m=(p), it's residue field k=A/m...now Hensel's lemma can be stated...

is it possible? I'm reading a proof where A is a local ring and f is a monic polynomial, v another polynomial in A[x] then the author says there is q,r with v=qf+r and degf<r. I thought the algorithm required divison of coefficients?! Maybe it's true we can always do this with f monic? thanks

thanks! I had that exact isomorphism in mind though I abandoned it because for some reason or another I thought it was false! Now however it's clear that k[x^2,x^3]=R is a k-algebra with 2 generators and thus there is an induced map from k[t,u] onto R. The kernel most certainly includes this...

I need help. So the obvious answer is no, because it's not integrally closed (incidentally it's integral closure is in fact k[x]. Here obviously k is a field. But I want to show that it is both noetherian and dimension 1 (nonzero prime ideals are maximal), here is my idea for a proof: (i...

I'm a high school senior who's never taken physics but I plan on doing so in college. I'd like to get a good start...I'm very good w/ the sciences..i'm in the IB-program so I really would like to get a good headstart..plan on studying over the summer. Just two days ago I bought PHYSICS...

hahah..cool. Thanks a bunch Chen! Just started this stuff yesterday (self-teaching)..using the book: PHYSICS: GIANCOLI (5th ed) - some tough problems. Any chance you could help me w/ the 2nd problem. Ok..ill just ask. the velocities of each component are defined as vcosx or vsinx right? Would it...

ok for the first problem: y-component data: the final y-velocity of bomb = squareroot(2(9.8)(88)) assuming initial velocity is zero. x-velocity is constant (195000/3600)m/s since y-velocity is opposite the desired angle and x-velocity is adjacent. tanx = -41.53/51.38 x = 51.05...