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here is the hw and handout

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
)