Math and stuff

Proposition Let $A$ be any ring, and consider the polynomial ring $A[T]$. Prove that $T$ is not a zero divisor in $A[T]$. Generalize the argument to prove that a monic polynomial \[\begin{align*} f = T^n + a_{n - 1}T^{n - 1} + \cdots + a_0 \end{align*}\] is not a zero...

What $x$ means $x$ can be a function. For instance, $x$ might mean $(x, y) \mapsto x$.

Example Let $F(x, y, z) = xy - z^2 \in \mathbb{C}[x, y, z]$. Since $\mathbb{C}$ is a field, it is a UFD. By Theorem 7 on P.304 of Dummit and Foote, $\mathbb{C}[x]$ is a UFD. By applying the same theorem repeatedly, we have that $\mathbb{C}[x, y, z]$ is a UFD....

Proposition Let $X = \mathbb{C}^n$ and $A = \mathbb{C}[x_1, \cdots, x_n]$. Maximal ideals of $A$ are in one-to-one correspondence with points $P \in X$. That is \[\begin{align*} P = (a_1, \cdots, a_n) \in X \iff m_P = (x_1 - a_1, \cdots, x_n - a_n) \subset A. \end{align*}\] Proof Unfortunately, I...

Simplified definition Let $F \in \mathbb{C}[x_1, x_2]$ be given. Then the locus $X = V(F) = \{ (a, b) \in \mathbb{C}^2 \mid F(a, b) = 0 \}$ is a hypersurface. This is not the correct definition, but this is sufficient for this post. See P.3 of Undergraduate Commutative Algebra for...