Here's the part about the kernel. First recall that if $A$ is a domain, $A[y]$ is the polynomial ring in the indeterminate $y$, and $f \in A[y]$ is a monic polynomial, then you can do Euclidean division by $f$ in $A[y]$. Apply this to $A = k[x]$, and $f = y - x^{2}$. If $g \in \ker(\phi)$, divide...

I am having trouble with this problem. We have to determine whether the set of integers such that $n!+1$ divides $(2012n)!$ is finite or infinite. Basically we have to determine if the prime factors of $n!+1$ are small enough for a finite or infinite number of integers, but I have not been able t...

Let $S=\sum\limits_{i=1}^n A_i$. We have $A_iS=S$ for all $i$. To see this notice $A_iS = (A_iA_1+ \dots + A_iA_n)$, and because the set of $A_i$ is a group this just gives us a sum for the elements of the group in a different order. In particular $S^2 = A_1S + \dots + A_nS = nS$. It follows $S^2...

Let $S=\sum\limits_{i=1}^n A_i$. We have $A_iS=S$ for all $i$. To see this notice $A_iS = (A_iA_1+ \dots + A_iA_n)$, and because the set of $A_i$ is a group this just gives us a sum for the elements of the group in a different order. In particular $S^2 = A_1S + \dots + A_nS = nS$. It follows $S^2...

Let $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is a group with matrix multiplication. If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ then how to prove $\sum _{i=1}^k M_i=0$ ?