Rings and Polynomials
Read this topicExplanations 3
- This is a stub article; the stuff on transformations and completing the square should be expanded. The goal of this post is to show (algebraically) that all quadratics have the …
By Victor Wang
- It is often thought that the imaginary unit iii, and complex numbers by extension, were created by "dreaming up" a solution to the equation x2=−1x^2 = -1x2=−1, for the purposes …
By Victor Wang
- Stub explanation The quadratic discriminant b2−4acb^2-4acb2−4ac of a real quadratic P(x)=ax2+bx+cP(x) = ax^2+bx+cP(x)=ax2+bx+c has a nice interpretation in terms of the roots r1…
By Victor Wang
Problems 4
- For which triples (R,f,g)(R,f,g)(R,f,g) below is the ring homomorphism R[x]/(fg)→R[x]/(f)×R[x]/(g)R[x]/(fg) \to R[x]/(f)\times R[x]/(g)R[x]/(fg)→R[x]/(f)×R[x]/(g) an isomorphism…
- Can we find a sequence of reals ai≠0a_i\neq0ai=0 such that each polynomial pn(x)=a0+a1x+⋯+anxnp_n(x) = a_0 + a_1 x + \dots + a_n x^npn(x)=a0+a1x+⋯+anxn has all its roots r…
- What is the asymptotic growth of I(n)I(n)I(n) as n→∞n\to\inftyn→∞? \\ [Source: Most recently communicated by Mark Sellke]
- How could you prove this identity?