MAT 150C University of California Davis Modern Algebra Questions Answers

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MAT 150C: MODERN ALGEBRA
Homework 3
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) = x2 + 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 xn −m is irreducible
in Q[x] for every n ≥ 1.
3. For n > 0, denote
n
e n (x) := x − 1 = xn−1 + xn−2 + · · · + x + 1 ∈ Z[x]
Φ
x−1
e p (x) is irreducible in Z[x] (equivalently, in Q[x]).
we have seen in class that if p is prime, then Φ
e m (x) divides Φ
e n (x) in Z[x].
(a) Assume m divides n. Show that Φ
e n (x) is irreducible in Z[x] if and only if n is prime.
(b) Conclude that Φ
e 2p (x) into irreducible factors in Z[x].
(c) Let p be an odd prime number. Decompose Φ
e n (x)
In general, we define the n-th cyclotomic polynomial Φn (x) inductively as follows. Decompose Φ
into irreducible, monic polynomials in Z[x]. One of this factors is not equal to Φm (x) for m < n. This e p (x) for p prime. In part (c) you have found a irreducible factor is Φn (x). For example, Φp (x) = Φ 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 (x − e 1≤d≤n gcd(d,n)=1 1 2πid n )