Recent content by joebohr

I seem to have figured out how to factor mod p (in a prime field) between a couple documents: "www.science.unitn.it/~degraaf/compalg/polfact.pdf" http://www.math.uiuc.edu/~r-ash/Ant/AntChapter4.pdf However, I'm still wondering what other types of finite fields it would be useful to...

I can understand most of Galois Theory and Number Theory dealing with factorization and extension fields, but I always run into problems that involve factorization mod p, which I can't seem to figure out how to do. I can't find any notes anywhere either, so I was wondering if someone could give...

A germ is essentially just a local topological structure.

How would this change if I were factoring over another extension field, say Q(√i)? Sorry for asking so many difficult questions. Could someone at summarize the results in the book or show how to solve this problem for an easier field? Thanks.

Ok, this clears some things up. I was wondering how you could factor over Q(√2) but this seems to explain the basics. However, I am still confused as to how you could solve p=x^2+ny^2 to determine if it is factorable.

I had no idea that the problem was so difficult or that it has not yet been solved. That pdf you posted a link to was great and very interesting, though. It seems like a very interesting field. I wonder which specific topics have been worked out within it. Furthermore, how can the method used by...

Wow, this is great work. My only questions are: how did you come up with this and how can it be applied to the problem at hand?

Well, I did some research, and according to wikipedia: "As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial x8 − 72x6 + 180x4 − 144x2 + 36...

Oops, I meant to ask "Why don't we find a problem like these two examples that has the quaternion group as its Galois Group." I know these are not the same things

Ok, if we started at a = 1/10 (8p1-7p2+9p3) b = 1/10 (9p2-6p1-13p3) and c = 1/10 (3p2-2p1-p3) Then what would these congruences look like and how many solutions would they have? Ultimately, we're going to get a large number (the sum of six numbers each that represent the number of solutions...

Yes, that is exactly what I meant. Ok, that explains why none of the texts I've read deal with Galois Groups over R. How about we make it interesting and try to find the Galois Groups of the first two examples over the quaternions!

I don't know why I put a 12 either; you're right, it's obviously supposed to be a 4. The GCD would still be 1, though. More info about Florentin Smarandache and Linear Bialgebra...

I was just talking about the real numbers for my base field. I'm not sure about what you mean by a "trivial" Galois Group, and I'm mostly interested in computing the Galois Group without solving the polynomial, but would the Galois Group of the first example just be isomorphic to the group of...

What I meant (and I was probably using incorrect terminology) was that the congruence has 1 solution that is less than 12.