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Math 246a, Homework 1
Christoph Thiele
due April 13
Problem 1
a) prove that 2 has no square root in Q. a) Without using trigonometric or exponential functions, and without appealing to the fundamental theroem of algebra, prove that every complex number has a square root. Where do you use completeness of R? b) Prove that 3 does not have a square root in Q +
2 Q. Does 1 +
2 have a square root in Q +
2 Q?
Problem 2
The Gaussian integers are the complex numbers of the form a + ib with integers a and b. a) Determine all Gaussian integers whose inverse is again a Gaussian integer (in- vertible elements) b) A Gaussian prime number z is a non-invertible element such that for every factorization z = z 1 z 2 into Gaussian integers either z 1 or z 2 is invertible. List all prime numbers of modulus less than 10 in the sector <(z) > 0, −<(z) < =(z) ≤ <(z) (why this sector?). c) Make a conjecture which primes in Z are Gaussian primes.
Problem 3
Let T be a non-zero linear transformation of R^2 given in the natural way by a 2 × 2 matrix. Prove that T preserves angles, i.e.,
〈x, y〉 ‖x‖‖y‖
〈T x, T y〉 ‖T x‖‖T y‖
for all x, y ∈ R^2 , if and only if the matrix of T is of the form ( a b −b a
) or
( a b b −a
)
for some real numbers a and b. What is the geometric difference between the two types?
Problem 4
The quaternion algebra is a (non-commutative) algebra which as a vector space is R^4 and has the multiplication
(a, b, c, d)(a′, b′, c′, d′) =
= (aa′^ − bb′^ − cc′^ − dd′, ab′^ + ba′^ + cd′^ − dc′, ac′^ + ca′^ + db′^ − bd′, ad′^ + da′^ + bc′^ − cb′)
Identify a subalgebra of the algebra of all 2 × 2 complex matrices which is iso- morphic to the quaternion algebra, and give the isomorphism. (This is a way to establish that indeed the above product defines an algebra). Use the matrix algebra representation to show that every element x in the quaternion algebra has an inverse element y, i.e., yx = xy is the identity element.
Problem 5
Prove that for every complex 2 × 2 matrix A there is a complex number λ and a vector v ∈ C^2 such that Av = λv
Do not just quote the corresponding theorem from linear algebra.
Problem 6
a) Assume that C is a circle in the complex plane C not containing 0. Prove that { 1 /z : z ∈ C} is again a circle. b) More generally, consider a linear fractional transformation
f (z) =
az + b cz + d
Let C be some circle in the complex plane. Prove that the set
{f (z) : z ∈ C}
is again a circle in the plane, provided cz + d 6 = 0 for all z ∈ C. (One possibility is to prove this first for certain special cases of a,b,c,d, and then deduce the general case by writing the map f as a composition of maps of the special types) c) What happens if cz + d = 0 for some z ∈ C? d) For which values of a, b, c, d is the unit circle C = {z : |z| = 1} mapped to itself?
