Mark each of the following true or false. 

__ a. Every Euclidean domain is a PID. 

__ b. Every PID is a Euclidean domain. 

__ c. Every Euclidean domain is a UFD. 

__ d. Every UFD is a Euclidean domain. 

__ e. A gcd of 2 and 3 in Q is ½

__ f. The Euclidean algorithm gives a constructive method for finding a gcd of two integers. 

__ g. If v is a Euclidean norm on a Euclidean domain D, then v(1) ≤ v( a) for all nonzero a ∈ D.

___ h. If v is a Euclidean norm on a Euclidean domain D, then v(1) < v(a) for all nonzero a ∈ D, a ≠ 1. 

___ i. If v is a Euclidean norm on a Euclidean domain D, then v(1) < v(a) for all nonzero nonunits a ∈ D. 

___ j. For any field F, F[x] is a Euclidean domain.