6.4.13. If R is the ring of Gaussian integers, show that Q(R) is
isomorphic
to the...
6.4.13. If R is the ring of Gaussian integers, show that Q(R) is
isomorphic
to the subfield of C consisting of complex numbers with rational
real and
imaginary parts.
3. Show that the Galois group of (x2 −
3)(x2 + 3) over Q is isomorphic to Z2 ×
Z2.
4. Let p(x) be an irreducible polynomial of degree n over a
finite field K. Show that its Galois group over K is cyclic of
order n.
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.
d) N/f(M) satisfies the Universal Property of Cokernels.
Q2. Show that ZQ :a) contains no minimal Z-submodule
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.
d) N/f(M) satisfies the Universal Property of Cokernels.