/sci/ - Science & Math

Autism = Poor social skills

Depression = Shitty life

Anxiety = Too many stimulants/Depressant withdrawl

Prove that the fraction [math]\displaystyle \frac{21n+4}{14n+3}[/math] is irreducible for every natural number [math]n[/math].

bonus round
*flips to a random page*

Let [math]p[/math] be a prime and let [math]P[/math] be a Sylow [math]p[/math]-subgroup of the finite group [math]G[/math]. Show that for any [math]G[/math]-module [math]A[/math] and all [math]n\geq0[/math] the map [math]\text{Res} : H^{n}(G,A)\to H^{n}(P,A)[/math] is injective on the [math]p[/math]-primary component of [math]H^{1}(G,A)[/math]. Deduce that if [math]|A|=p^{a}[/math] then the restriction map is injective on [math]H^{n}(G,A)[/math].

Bonus round:

Let [math]A\in M_{m,n}(\mathbb{F})[/math] and let [math]B\in M_{m+n,m+n}(\mathbb{F})[/math] be the square matrix [math]B = \begin{bmatrix} 0 & A \\ A^{*} & 0 \end{bmatrix}_{\text{block}} [/math]. Show that, counting multiplicity, the nonzero eigenvalues of [math]B[/math] are precisely the singular values of [math]A[/math] together with their negatives.

last one
*flips book to random page*

Let [math]R[/math] be a Noetherian ring, [math]I[/math] and ideal of [math]R[/math], and consider the Rees algebra [math]\text{Rees}_{R}(I) = \bigoplus_{\ell\geq 0}I^{\ell}[/math]. It is known that [math]\text{Rees}_{R}(I)[/math] is Noetherian (check this).

>Part 1:
For [math]\ell\geq 0[/math], let [math]J_{\ell}\subseteq I^{\ell}[/math] be ideals of [math]R[/math], and view [math]J := \bigoplus_{\ell\geq 0} J_{\ell}[/math] as a sub-[math]R[/math]-module of [math]\text{Rees}_{R}(I)[/math]. Prove that [math]J[/math] is an ideal of [math]\text{Rees}_{R}(I)[/math] if and only if [math]I^{n}J_{\ell}\subseteq J_{\ell+n}[/math] for all [math]\ell,n \geq 0[/math].
>Part 2:
Assume [math]J := \bigoplus_{\ell\geq 0} J_{\ell}[/math] is an ideal of [math]\text{Rees}_{R}(I)[/math] . Prove that [math]J[/math] admits a finite set of homogeneous generators.
>Part 3:
Choose a finite set of homogeneous generators for [math]J[/math], and let [math]s[/math] be the largest degree of an element in this set. Prove that [math]J_{s+1}\subseteq I^{s+1}J_{0} + I^{s}J_{1} + \cdots + IJ_{s}[/math] (as ideals of [math]R[/math]).

Good luck.

oh, I should leave a problem that janny can do so he doesn't feel left out:

Find the roots of [math]p(x)=x^{2}-4x+3[/math].

What to study next after my pure math degree to make myself future-proof?