the blank: 1. Let F be a field equipped with a Euclidean valuation δ: D\{0} → N∪{0} defined by δ(f)=deg f. If δ(f)=0 then f(x) must be a ____. 2. Let f and g be nonzero polynomials in F[x]. Then the unique greatest common divisor of f and g is the polynomial p that divides both f and g such that p is ____ and of ____. 3. Let F[x] be the ring of polynomials with coefficients over a field F. We say that p is an irreducible polynomial in F[x] iff deg p > 0 and the factors of p must be a unit and its _____.

Fill in the blank:

1. Let F be a field equipped with a Euclidean valuation δ: D\{0} → N∪{0} defined by δ(f)=deg f. If δ(f)=0 then f(x) must be a ____.
2. Let f and g be nonzero polynomials in F[x]. Then the unique greatest common divisor of f and g is the polynomial p that divides both f and g such that p is ____ and of ____.
3. Let F[x] be the ring of polynomials with coefficients over a field F. We say that p is an irreducible polynomial in F[x] iff deg p > 0 and the factors of p must be a unit and its _____.