Answer to Question #106453 in Abstract Algebra for Garima Ahlawat

Question #106453
Show that d : Q[x]\{0} -> N U {0} : d(f) = 5^deg f is a Euclidean valuation on Q[x]
1
Expert's answer
2020-03-25T12:52:59-0400

We know that for every "A,B\\in\\mathbb Q[x]" there are "P,R\\in\\mathbb Q[x]" such that "A=BP+R", where "\\deg R<\\deg B" or "R=0". Then "R=0" or "d(R)=5^{\\deg R}<5^{\\deg B}=d(B)".


We obtain that for every "A,B\\in\\mathbb Q[x]" there are "P,R\\in\\mathbb Q[x]" such that "A=BP+R", where "d(R)<d(B)" or "R=0".


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