Recent content by rookandpawn

Right, but that's just verifying one instance. It goes back to what i was saying earlier, that i have to write the formal product g(x+c)h(x+c) (meaning if g(x) = b_0 + b_1x^1 + .. b_nx^n and h(x) = c_0 + ... + c_mx^m then g(x+c)h(x+c) = b_0c_0 + ... + b_nc_mx^{n+m} which if i actually wrote it...

isn't the factorization g(x)h(x) just for the polynomial described by f(x)? the polynomial f(x+c) is a fundamentally different polynomial? I realize that f evaluated at x+c can be thought of as g evaluated at x+c times h evaluted at x+c, but that's as far as i can get? just because the...

Homework Statement Let F be a field and f(x) in F[x]. If c in F and f(x+c) is irreducible, prove f(x) is irreducible in F[x]. (Hint: prove the contrapositive) Homework Equations So, I am going to prove if f(x) is reducible then f(x+c) is reducible. The Attempt at a Solution f(x)...

Posted on reddit. Does he translate a partial amount of vertical energy into horizontal energy? when he bends his knees, he seems to "push off" before he hits the ground. I can definitely believe he timed it, and that he has enough coordination / gusto to come up with a life-saving plan in...

and made a complete ass out of yourself by posing questions that showed your ignorance? They lied when they said there are no dumb questions. That's only true when you're in high school and aren't expected to know much anyways, but near the graduate university level, there is such a thing as...

Oh okay. Heh.

(2x + 1) = 2(x + 3) (3x + 1) = 3(x + 2) = 6(x + 3)(x + 2) = (x+3)(x+2) = x^2 + 5x + 6 = x^2 + 1 so x^2 + 1 = (x + 2)(x + 3).

Nevermind, they are all units in Z_5. I can't believe i overlooked this. x^2 + 2 = (ax + b) (a^-1x + c) = a(x + b/a) a^-1(x + ca) = (x + b/a)(x + ca).

Homework Statement Show x^2 + 2 in Z_5[x] is irreducible. This is before the section on the factor theorem (j is a root -> (x-j) is a factor). So I'm not so sure I want to start checking for zero's since "its not available" per se. Homework Equations The Attempt at a Solution...

Oh haha.. My mistake.. I am just not thinking, x is a 2x1 column vector. So you did \left(\begin{array}{cc}a&b\\c&d\end{array}\right) \left(\begin{array}{c}e\\f\end{array}\right) = \left(\begin{array}{c}0\\0\end{array}\right) and then solve for e and f, and by inspection that e=dand f = -c?

Earlier, in reply #2 when I posted my attempted solution, isn't the process of finding A^{-1} through the gaussian elimination going to net me both directions. I.e. once I find that A^{-1} = \left(\begin{array}{cc}d/(ad-bc)&b/(ad-bc)\\-c/(ad-bc)&a/(ad-bc)\end{array}\right), that nets me that if...

First, thank you for your reply. I appreciate it because you brought up a point that I had tried to understand but wasn't totally conscious of, but knew there was something bad lurking in the background. That is, if A = (a b | c d) is a unit, there is SOME element B s.t. AB = I = BA. I...

My attempted solution so far: Prove that (a b | c d) is a unit in the ring M(R) iff ad-bc != 0. -> A = (a b | c d) is a unit. hence there is an element U in M(R) s.t. AU = 1 = UA where 1 = (1 0 | 0 1). This element U can be found through Gaussian Elimination. We start with the...

Homework Statement Prove that (a b c d) is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is (d/t -b/t -c/t a/t) where t= ad-bc. Homework Equations An element a in a ring R with identity is called...

Homework Statement An element a of a ring is nilpotent if a^n = 0 for some positive integer n. Prove that R has no nonzero nilpotent elements if and only if 0 is the unique solution of the equation x^2 = 0 Homework Equations I think nilpotent means that not only that a^n = 0 for...