Let F be a field, phi a valuation of F, and sigma an automorphism of F. Show that the composition phi\circ sigma is a valuation of F.
Show that -1 is the square of a p-adic integer when p = 5 but not when p = 7.
Show that any irreducible element of a principal ideal domain A gives rise to a non-trivial valuation of the fraction field of A in a way that is analogous to the way that a prime integer gives rise to a non-trivial valuation of Q.
Review of the universal mapping property for quotients: If R is a ring and I is a 2-sided ideal in R, an abstract quotient of R modulo I is a ring homomorphism pi : R -----> A with I <= Ker(pi) satisfying the condition that if f : R -----> S is any ring homomorphism with I <= Ker(f), then there is a unique ring homomorphism lambda : A -----> S such that lambda\circ pi = f .
Of course, if A = R/I is the set of additive cosets of I in R and pi : R -----> A is the map that assigns to each element of R its additive coset modulo I, then pi is an abstract quotient of R modulo I.
Prove that if pi : R -----> A is any abstract quotient of R modulo I, then the unique ring homomorphism lambda : R/I -----> A arising from the fact that R/I is an abstract quotient is the inverse of the unique ring homomorphism mu : A -----> R/I arising from the fact that A is an abstract quotient of R modulo I.
Prove that if pi is an abstract quotient of R modulo I, then pi is surjective.
If phi is a non-Archimedean valuation of a field F, then phi induces a group homomorphism F^{*} -----> R^{+} from the multiplicative group of F to the multiplicative group of positive real numbers. One calls phi(F^{*}) the value group of phi, and one says that phi is a discrete valuation if its value group is a discrete subgroup of R^{+}.
Show that the completion of a discrete valuation is a discrete valuation having the same value group.