The obvious place to start is with a rigorous definition of the ring of polynomials over a ring R. Definition 22.1. Let R be a ring. A polynomial f(x) with indeterminate x and coefficients in R is a formal sum f(x) = ¥ å k=0 a kx k= a 0 + a 1x + + a kx + where all but finitely many of the coefficients a k 2R are non-zero. The degree of f ...
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Polynomial Rings
Definition. Let R be a commutative ring and let f(x) ∈ R[x]. An element c ∈ R is a root of f(x) if f(c) = 0. Note that polynomials are actually formal sums, not functions. However, it is obvious how to plug a number into a polynomial. Specifically, let f(x) = a nxn +a n−1xn−1 +···+a1x+a0 ∈ R[x]. For c∈ R, define f(c) = a ncn +a
We start with some basic facts about polynomial rings. Lemma 21.1. Let Rbe an integral domain. Then the units in R[x] are precisely the units in R. Proof. One direction is clear. A unit in Ris a unit in R[x]. Now suppose that f(x) is a unit in R[x]. Given a polynomial g, denote by d(g) the degree of g(x) (note that we are not claiming that
Polynomial rings give interesting examples of in nite rings of nite characteristic. For example Z 2[x] has in nitely many polynomials| just let the degree go to in nity|but the characteristic is two. Indeed if you add a polynomials to itself you are just adding the coe cients to themselves, which are then all zero. More generally Z
Chapter 12. Polynomial Rings 12.1 Note: Here are a few remarks about polynomials. Recall that R[x] denotes the ring of polynomials with coe cients in the ring R, and RR denotes the ring of all functions f : R !R. (1) A polynomial f 2R[x] determines a function f 2RR. Given f(x) = Pn i=0 a ixi 2R[x] we obtain the function f : R !R given by f(x ...
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. Proof. One direction is clear. A unit in R is a unit in R[x]. Now suppose that f(x) is a unit in R[x]. Given a polynomial g, denote by d(g) the degree of g(x) (note that we are not claiming that
Polynomial Rings 1. Definitions and Basic Properties For convenience, the ring will always be a commutative ring with identity. Basic Properties The polynomial ring R[x] in the indeterminate xwith coe cients from Ris the set of all formal sums a nxn + a n 1xn 1 + + a 1x+ a 0 with n 0 and each a i 2R. Addition of polynomials is componentwise: Xn ...
Polynomial Rings De nition: Let R be a commutative ring with 1. The polynomial ring R[x] in indeterminate x with coe cients from R is the set of all formal sums a nxn + a n 1xn 1 + + a 1x + a 0 with n 0 and a i 2R. That is, R[x] = fa nxn + a n 1xn 1 + + a 1x + a 0;n 0;a i 2Rg. In order to make a ring out of R[x] we must be able to recognize
In this section we study the ring of polynomials in n variables x1, . . . , xn with coefficients in our field K. This polynomial ring is denoted by K[x] = K[x1, . . . , xn]. If the number n is small, then we typically use letters without indices to denote the variables.
Polynomilas are formal expressions and (in algebra) they are manipulated formally. De ̄nition 2.1 Let F be a ̄eld and N = f0; 1; 2; : : :g be the set of non-negative integers. Let F denote the set of all functions f : N ! F: So, is the set of all in ̄nite sequences in F: De ̄ne addition and multiplication on F naturally (see the book).