Unit of polynomial ring：Characterizing units in polynomial rings
Characterizing units in polynomial rings
18.703 Modern Algebra
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We start with some basic facts about polynomial rings. Lemma 21.1. Let R be an integral domain. Then the units in R[x] are precisely the units in R.
group theory
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Where Z[x] is the ring of polynomials in x with integer coefficients. The book I am studying says the unity of this ring is f(x)= ...
Identifying units in a polynomial ring
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First, prove that, given two non-zero polynomials, the degree of fg is the degree of f + the degree of g. [Here you'll need that R is a domain ...
Nilpotents
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A polynomial in A[x] is nilpotent if and only if all of its coefficients are nilpotent in A. Proof. The nilpotent elements in a commutative ring form an ideal, ...
Polynomial ring
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In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more ...
Unit (ring theory)
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A nonzero ring R in which every nonzero element is a unit (that is, R× = R ∖ 0}) is called a division ring (or a skew-field). A commutative division ring is ...
Units in Polynomial Rings
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A polynomial is a unit if its constant term is a unit, and the coefficients of its higher degree terms are nilpotent.
Units of Ring of Polynomial Forms over Field
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Theorem. Let (F,+,∘) be a field whose zero is 0F and whose unity is 1F. Let F[X] be the ring of polynomial forms in an indeterminate X over ...