HOMEWORK 9 — DUE NOV 21ST

MATH 25

There are two sections: A and B. Please return each part to the proper CA. In all these problems, if R is a ring, R^{∗} = R \ {0}.

A. Problems graded by Benjamin

A.1. Let (E, d) be a metric space, and let f : E → C be a function. Write f(x) = a(x) + ib(x) where a, b are two functions from E to R.

(1) prove that f is continuous if and only if a and b are continuous.

(2) prove that if f, g are two continuous functions from (E, d) to C then f ± g and f · g are also continuous.

A.2. If m ∈ Z_{≥1} let ϕ(m) = card{1 ≤ x ≤ m, gcd(x, m) = 1}. If G is a group and if g ∈ G, the order of g is the smallest m ≥ 1 such that g^{m} = e. Don’t confuse this with the order of G which is card G.

(1) prove that the order of g is the order of the subgroup it generates in G (assume that G is finite).

(2) prove that if G is any finite group and if g ∈ G then the order of g divides the order of G.

(3) prove that if n ≥ 1, then ∑_{d|n} ϕ(d) = n (here ∑_{d|n} means the sum over all divisors d of n).

(4) show that if G is a cyclic group of order n and d divides n, then there are ϕ(d) elements in g of order d.

(5) in (5)-(7) let K be a field and let G be a finite subgroup of (K^{∗}, ×) of order n. Prove that there are at most d elements g ∈ G such that g^{d} = 1.

(6) prove that there exists an element g ∈ G which is of order n.

(7) prove that G is cyclic.

(8) let G be the set of invertible elements in Z/24Z. Prove that G is not a cyclic group.

A.3. Let S^{1} be the set of complex numbers z such that |z| = 1 and let C_{n} be the set of complex numbers such that z^{n} = 1.

2 MATH 25

(1) prove that (S^{1}, ×) and (C_{n}, ×) are compact subgroups of (C^{∗}, ×).

(2) are there any other ones?

A.4. If P (X) = X^{d} + a_{d−1}X^{d−1} + ⋯ + a_{0} ∈ C[X], let R be the largest of the |z_{i}| where z_{1}, ⋯, z_{d} are the roots of P in C. Prove the following inequalities:

(1) if r > 0 satisfies r^{d} ≥ ∑_{i=0}^{d−1} |a_{i}|r^{i}, then R ≤ r.

(2) R ≤ max(1, ∑_{i=1}^{d−1} |a_{i}|) (Montel)

(3) R ≤ 1 + max_{1≤k≤d} |a_{k}| (Cauchy)

(4) R ≤ |1 − a_{d−1}| + |a_{d−1} − a_{d−2}| + ⋯ + |a_{1} − a_{0}| + |a_{0}| (Montel)

(5) if a_{i} ∈ R for all i and 1 ≥ a_{d−1} ≥ a_{d−2} ≥ ⋯ ≥ a_{0} ≥ 0 then R ≤ 1 (Kakeya)

B. Problems graded by Inna

B.1. Find a formula for ∑_{k=0}^{n} cos(kx) and for ∑_{k=0}^{n} sin(kx). Hint: consider 1+e^{ix}+⋯+e^{inx}.

B.2. Prove that if P ∈ C[X] is a non-constant polynomial, then the function z ↦ |P (z)| has no local maximum in C (a local maximum is a point z_{0} ∈ C such that there exists δ > 0 with the property: if |z − z_{0}| < δ, then |P (z)| ≤ |P (z_{0})|).

B.3. Let P (X) = a_{0} + a_{1}X + ⋯ + a_{d}X^{d} ∈ R[X] be a polynomial of degree d. We say that P is split in R if one can write P (X) = a_{d}(X − α_{1}) ⋯ (X − α_{d}) where the α_{i}’s are (not necessarily distinct) elements of R. For example X^{2} − 2X + 1 is split but not X^{2} + 1.

(1) prove that if P is split then P′ is split.

(2) more generally, prove that if λ ∈ R, and if P is split, then P′ + λP is split (hint: consider the function x ↦ P (x)e^{λx}).

(3) prove that if a polynomial b_{0} + b_{1}X + ⋯ + b_{d}X^{d} and P (X) are both split then so is the polynomial b_{0}P (X) + b_{1}P′(X) + ⋯ + b_{d}P^{(d)}(X).

B.4. If z_{1}, ⋯, z_{d} are complex numbers, the convex hull of the z_{i}’s is the set of complex numbers of the form z = λ_{1}z_{1} + ⋯ + λ_{d}z_{d} where λ_{i} ∈ R_{≥0} and λ_{1} + ⋯ + λ_{d} = 1.

(1) for n = 2, 3, 4, 5 describe the convex hull of the n solutions of the equation z^{n} = 1 (draw a picture!).

(2) Let P ∈ C[X] be a polynomial of degree d, whose roots are z_{1}, ⋯, z_{d}. Prove that

\frac{P′(X)}{P (X)} = ∑_{i=1}^{d} \frac{1}{X − z_{i}}

(3) use this to show that the roots of P′(X) lie in the convex hull of the roots of P (X).