EMandatory exercises AC24

Let $p$ be a prime number and $k$ a field containing a primitive $2p$-th root of unity $\zeta$. You may assume that $k=\mathbb{C}$ below. If you have time, you can adjust the exact conditions on the field.

1. Prove that
   $$t^p + 1 = (t - \zeta) (t - \zeta^3) \cdots (t - \zeta^{2p-1})\in k[t].$$ Show that this implies
   $$x^p + y^p = (x - \zeta y) (x - \zeta^3 y) \cdots (x - \zeta^{2p-1} y) \in k[x, y].$$
2. Suppose that $f, g\in k[t]$ with $\gcd(f, g) = 1$. Show that
   $$\gcd(f - \zeta^i g, f - \zeta^j g) = 1$$ for $i, j = 1, \dots, 2p-1$ and $i\neq j$.
3. Suppose that $p\geq 3$ and $f^p + g^p = h^p$ for non-constant polynomials $f, g, h\in k[t]$ with $\gcd(f, g) =1$. Show that
   $$\begin{aligned} f - \zeta g &= a^p\\ f - \zeta^3 g &= b^p\\ f - \zeta^5 g &= c^p \end{aligned}$$ for suitable polynomials $a, b, c\in k[t]$.
4. Show that $a^p, b^p$ and $c^p$ are linearly dependent over $k$. Use this and induction on $\deg(f) + \deg(g)$ to prove Fermat's last theorem for function fields: there does not exist non-constant polynomials $f, g, h\in k[t]$ with $\gcd(f, g) = 1$, such that
   $$f^n + g^n = h^n$$ for $n\geq 3$.

Consider the two plane curves $V(f)$ and $V(g)$ in $\mathbb{C}^2$, where $$\begin{aligned} f &= y^2 - x^3 + x\\ g &= y^3 - x^2. \end{aligned}$$ Let $I$ denote the ideal $(f, g)$ in $R = \mathbb{C}[x, y]$.

1. Show that $f$ and $g$ do not intersect at $\infty$.
2. Show that $I$ is not a radical ideal.
3. Show that $R/I$ is a finite dimensional vector space over $\mathbb{C}$ and that $\dim_\mathbb{C} R/I = 9$ by explicitly constructing a basis for $R/I$.
4. Let $P = (0, 0)\in V(f, g)$. Compute
   $$\dim_\mathbb{C} R_P/I_P$$
   Prove that

   $$I = \left(y^3 - y^4 - 2y^6 + y^9, x + y^2 + y^4 - y^7 \right)$$ and work with those generators.
   and interpret your result comparing with the output of

In this exercise we consider $X = \mathbb{P}^2(\mathbb{C}) = \{[(x_0, x_1, x_2)] \mid (x_0, x_1, x_2)\in \mathbb{C}^3\setminus\{0\}\}$ with the euclidean topology induced from $\mathbb{C}^3\setminus\{0\}$. Let $$\begin{aligned} f &= x_1^2 + x_2^2 - 1\\ F &= x_1^2 + x_2^2 - x_0^2\\ A &= \{[(1, x, y)] \mid f(x, y) = 0\} \subset X \end{aligned}$$

1. Show that
   $$V(F) = \{[(x_0, x_1, x_2)] \mid F(x_0, x_1, x_2) = 0\}$$ is well defined and closed in $X$.
2. Show that $A\subset V(F)$ and $\mid V(F)\setminus A \mid = 2$. Find these two points!
3. Prove that $\overline{A} = V(F)$ in $X$.
4. A line in $X$ is a subset of the form $L = \{[(x_0, x_1, x_2)] | H(x_0, x_1, x_2) = 0\}$, where $H = a x_0 + b x_1 + c x_2$ and $(a, b, c) \neq (0, 0, 0)$. Given finitely many points
   $$P_1, \dots, P_r\in X,$$ prove that there exists a line $L$ not containing any of these points.