Fall 2013
Math 581
Problems 1
09/24/2013
1) Let H be a group. Denote by Aut(H) the set of group isomorphisms of H. This is a
group with composition of maps as multiplication. Let further G be a group and
: G Aut(H)
a homomorphism of groups. We set g h :=
Calculus is concerned with rates of change.
Examples:
-velocity and density, current, temperature gradient in physics;
-rate of reaction and compressibility in chemistry;
-rate of population growth and blood velocity in biology;
-marginal cost and margina
-On any vector space we have the identification of the tangent space at any
point with the vector space itself.
-In particular, with have the vector field E which assigns to each point p
itself, thought of as an element of the tangent space at p. In terms
-Important in differential calculus
-Slope of a line through points (x1, y1) and (x2, y2) is m = y2y1/x2x1
-Point-slope form: y y0 = m(x x0)
-Slope-intercept form: y = mx + b
-Horizontal lines: y = k
-Vertical lines: x = h
-Parallel lines: same slopes
-Pe
Fall 2013
Math581
Solutions to Problems 1
1) It is a straightforward calculation to show that the product is associative,
(eH , eG ) is the neutral element, where eH and eG denote the neutral ele1
ments of H and G, respectively, and the inverse of (x, g)
Fall 2013
Math581
Solutions to Problems 3
1) We have a K-algebra isomorphism
: K(a) K[T ]/(f (T ) ,
which maps a to the class of T (here (f (T ) K[T ] is the ideal generated
by f (T ). Hence it is enough to show that there is a K-algebra homomorphism K[T
Fall 2013
Math581
Solutions to Problems 5
1) Let = 3 5 R a positive 3rd root of 5. Then all roots of T 3 5 are
given by , , and 2 , where = 3 C is a 3rd root of unity. Hence
L = Q(, ) and Gal(L/Q)
S3 . The group S3 has three subgroups of
order 2 which cor
Fall 2013
Math581
Solutions to Problems 4
1) If l = p d for some integer d 1 then
T l 1 = (T d )p 1 = (T d 1)p
(the last equation since char F = p). Hence T l 1 is not separable.
If p does not divide l then f (T ) = l T l1 = 0 and has 0 as only root.
But
Fall 2013
Math 581
Problems 3
10/22/2013
1) Let K be a eld, K a eld extension which is algebraically closed, and K(a) K
a simple algebraic eld extension. Let further f (T ) K[T ] be the minimal polynomial
of a. Show that if b is a root of f (T ) in then t
-All possible values of k can occur as the constant curvature. Indeed
spheres have constant positive curvature, Euclidean spaces have k = 0
and k < 0 in hyperbolic geometry. In all of these cases, the sectional
curvature is not only constant on the two pl