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... Theorem. The polynomial ring over a field is a Euclidean domain. Proof. Let K[X] be the polynomial ring over a field K in the indeterminate X. Since K is an integral domain and any polynomial ring over integral domain is an integral domain, the ring K[X] is an integral domain. The degree ν(f ), defi ...
... Theorem. The polynomial ring over a field is a Euclidean domain. Proof. Let K[X] be the polynomial ring over a field K in the indeterminate X. Since K is an integral domain and any polynomial ring over integral domain is an integral domain, the ring K[X] is an integral domain. The degree ν(f ), defi ...
Math 322, Fall Term 2011 Final Exam
... algebraic? Give a necessary and sufficient condition for an element α in E to be algebraic over F . Give an example of a complex number which is not real and is algebraic over Q. ...
... algebraic? Give a necessary and sufficient condition for an element α in E to be algebraic over F . Give an example of a complex number which is not real and is algebraic over Q. ...
Grobner
... • Many algorithmic problems can be solved easily with this basis. • One example (focus of our lecture): abstract ideal membership problem: – Is a given polynomial g in a given ideal I ? I { f1 , f r } – Equivalently: can g be expressed as an algebraic combination of the fj for some polynomials hj ...
... • Many algorithmic problems can be solved easily with this basis. • One example (focus of our lecture): abstract ideal membership problem: – Is a given polynomial g in a given ideal I ? I { f1 , f r } – Equivalently: can g be expressed as an algebraic combination of the fj for some polynomials hj ...