MODERN ALGEBRA

Instructions. Please write the answer to each problem, including the computational ones, in connected
sentences and explain your work. Just the answer (correct or not) is not enough. Write your name in every
page and upload to Gradescope with the correct orientation. Make sure to indicate to Gradescope which
pages correspond to each problen.

1. Recall that SL2(C) is the group of all 2 × 2 complex matrices with determinant 1. In this problem, we
   will show that SL2(C) has the structure of an irreducible algebraic variety.
   (a) Let f(x, y) = x
   2 + y
   2 − 1 ∈ C[x, y]. Show that V (f) does not contain any line of the form
   L = V (ax + by + c). Conclude that f is irreducible.
   (b) Now let g(x, y, z, w) = xy − zw − 1 ∈ C[x, y, z, w]. Use part (a) to show that g is irreducible.
   Explain why this implies that SL2(C) is an irreducible algebraic variety.
   (c) Show that h(x, y, z, w) = xy − zw ∈ C[x, y, z, w] is irreducible. (Note: this does not follow
   immediately from (b))
2. Let m ∈ Z be a square-free number (meaning that no square divides m). Show that x
   n−m is irreducible
   in Q[x] for every n ≥ 1.
3. For n > 0, denote
   Φen(x) := x
   n − 1
   x − 1
   = x
   n−1 + x
   n−2 + · · · + x + 1 ∈ Z[x]
   we have seen in class that if p is prime, then Φep(x) is irreducible in Z[x] (equivalently, in Q[x]).
   (a) Assume m divides n. Show that Φem(x) divides Φen(x) in Z[x].
   (b) Conclude that Φen(x) is irreducible in Z[x] if and only if n is prime.
   (c) Let p be an odd prime number. Decompose Φe2p(x) into irreducible factors in Z[x].
   In general, we define the n-th cyclotomic polynomial Φn(x) inductively as follows. Decompose Φen(x)
   into irreducible, monic polynomials in Z[x]. One of this factors is not equal to Φm(x) for m < n. This
   irreducible factor is Φn(x). For example, Φp(x) = Φep(x) for p prime. In part (c) you have found a
   formula for Φ2p(x) for an odd prime p.
   In general, it is very hard to compute Φn(x). For example, if n has at most two prime factors then all
   the coefficients of Φn(x) are 0, +1 or −1. But Φ105(x) has a coefficient equal to −2 (and 105 is the
   smallest number where a coefficient distinct from 0, ±1 appears). If you like number theory, you can
   try showing that Φn(x) is a polynomial of degree Tot(n), where Tot is Euler’s totient function. If you
   like complex analysis you can try showing that
   Φn(x) = Y
   1≤d≤n
   gcd(d,n)=1
   (x