Answer to Question #138278 in Abstract Algebra for Raunaqe

Question #138278
Show that d:Q[x]\{0}→NU{0}:d(f)=2^deg f is a Euclidean valuation on Q[x].
1
Expert's answer
2020-10-14T14:41:13-0400

Let f(x),g(x)"\\in Q[x]"

Then there exists q(x) and r(x)"\\in Q[x]" such that f(x)=g(x)q(x)+r(x) where either r(x)=0 or deg(r)<deg(g). (Since Q is a field, division algorithm holds in Q[x])

When r"\\neq 0" , deg(r)<deg(g) "\\Rightarrow" 2"^ {deg(r)}" <2"^{deg(g)}" "\\Rightarrow d(r)<d(g)"

We also note d"(fg)=2^{deg(fg)}= 2^{deg(f)+deg(g)}=2^{deg(f)} 2^{deg(g)}=d(f)d(g)"

Hence d is an Euclidean evaluation.


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