Question

Show that d:QQ[x]\{0}→NN∪ {0}:d(f)=2^(deg f) is a Euclidean valuation on QQ[x].

Answer 1

Answer on Question #82539 – Math – Abstract Algebra

Question

Show that d:QQ[x]\{0\}→NN∪{0}:d(f)=2^(deg f) is a Euclidean valuation on QQ[x].

Solution

By the definition, we need to show that

1) d(f) <= d(fg);

2) ∀f, g ∈ Q[x] ∃q, r ∈ R: f = qg + r and either r=0 or d(r)<d(g).

1. deg(f) < deg(fg)=deg(f)+deg(g), hence d(f) = 2^deg(f) <= 2^deg(fg) = d(fg)

2. There is division with remainder in Q[x]

∀f, g ∈ Q[x] ∃q, r ∈ R: f = qg + r such that either r=0 or deg(r)<deg(g).

Thus, ∀f, g ∈ Q[x] ∃q, r ∈ R: f = qg + r such that either r=0 or

d(r) = 2^deg(r) < 2^deg(g) = d(g).

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