Abstract Algebra, Lecture 12 Jan Snellman Abstract Algebra, Lecture 12 Polynomial Rings

Abstract Algebra, Lecture 12 Jan Snellman Abstract Algebra, Lecture 12 Polynomial Rings

Abstract Algebra, Lecture 12 Jan Snellman Abstract Algebra, Lecture 12 Polynomial rings Polynomial rings Jan Snellman1 Coefficients in a domain 1Matematiska Institutionen Coefficients in a Link¨opingsUniversitet field Link¨oping,fall 2019 Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA55/ Abstract Algebra, Lecture 12 Jan Snellman Summary Division algorithm Polynomial rings 1 Polynomial rings K[x] is a PID Coefficients in a Zero-divisors, nilpotents, units GCD domain Degree Zeroes of polynomials and Coefficients in a Evaluation field linear factors 2 Coefficients in a domain Prime and maximal ideals in Divisibility K[x] Polynomial rings in several Unique factorization variables Ideal calculus in K[x] 3 Coefficients in a field Quotients Abstract Algebra, Lecture 12 Jan Snellman Summary Division algorithm Polynomial rings 1 Polynomial rings K[x] is a PID Coefficients in a Zero-divisors, nilpotents, units GCD domain Degree Zeroes of polynomials and Coefficients in a Evaluation field linear factors 2 Coefficients in a domain Prime and maximal ideals in Divisibility K[x] Polynomial rings in several Unique factorization variables Ideal calculus in K[x] 3 Coefficients in a field Quotients Abstract Algebra, Lecture 12 Jan Snellman Summary Division algorithm Polynomial rings 1 Polynomial rings K[x] is a PID Coefficients in a Zero-divisors, nilpotents, units GCD domain Degree Zeroes of polynomials and Coefficients in a Evaluation field linear factors 2 Coefficients in a domain Prime and maximal ideals in Divisibility K[x] Polynomial rings in several Unique factorization variables Ideal calculus in K[x] 3 Coefficients in a field Quotients Abstract Algebra, Lecture 12 Definition Jan Snellman Let L be a unitary, commutative ring. The polynomial ring L[x] is the set of all maps c : N L whose support Supp (c) = f n 2 c(n) 6= 0 g Polynomial rings ! N Zero-divisors, nilpotents, units is finite. Two such maps are added component-wise, and multiplied using Degree Evaluation Cauchy convolution Coefficients in a n domain (c ∗ d)(n) = c(i)d(n - i) Coefficients in a i=0 field X This makes L[x] into a commutative, unitary ring; via the injection L 3 ` 7 (N 3 n 7 ` 2 L) one can regard L as a subring.! ! Abstract Algebra, Lecture 12 Jan Snellman One usually uses the indeterminate x as a \placeholder" for the coefficients, so the map c : N L is displayed as Polynomial rings j Zero-divisors, f (!x) = c(j)x ; nilpotents, units 1 j=0 Degree X Evaluation j Coefficients in a where x can be thought of as the indicator function on fjg. domain The \Cauchy convolution" is then explained by the rule Coefficients in a field xi ∗ xj = xi+j and distributivity. Abstract Algebra, Lecture 12 Jan Snellman Example Polynomial rings Let L = Z, and let c : N L be given by c(0) = 2, c(1) = -3,c(2) = 1, Zero-divisors, and c(n) = 0 for n > 2. nilpotents, units Degree Let d : N L be given by!d(0) = -1, d(1) = 0,d(2) = 5, and d(n) = 0 Evaluation for n > 2. Coefficients in a domain The corresponding! polynomials, and their product, are Coefficients in a 2 2 2 3 4 field (2 - 3x + 1x ) ∗ (-1 + 0x + 5x ) = -2 + 3x + 9x - 15x + 5 ∗ x Abstract Algebra, Lecture 12 Jan Snellman Theorem L[x] is an integral domain iff L is. Proof. Polynomial rings Zero-divisors, If ab = 0 in L, then the same holds in L[x]. nilpotents, units Degree If Evaluation n m Coefficients in a 0 = (a0 + a1x + ··· + anx )(b0 + b1x + ··· + bmx ) domain n+m = a0b0 + (a0b1 + a1b0)x + ··· + anbmx Coefficients in a field then anbm = 0. Example Z[x] is a domain, Z6[x] is not. Abstract Algebra, Lecture 12 Jan Snellman Theorem n f = a0 + a1x ··· + anx 2 L[x] is invertible iff a0 is a unit in L and Polynomial rings a1;:::; an are nilpotent in L. Zero-divisors, nilpotents, units Degree Proof Evaluation We will make use of the fact (proved later) that Coefficients in a domain • nilpotent + nilpotent = nilpotent Coefficients in a • unit + nilpotent = unit field n If a0 unit, a1;:::; an nilpotent, then r = a1x + ··· + anx nilpotent, so f = a + r unit. Abstract Algebra, Lecture 12 Jan Snellman Proof (cont) n Conversely, if f = a0 + a1x + ··· + anx is a a unit, then there exists m g = b0 + b1x + ··· + bmx with fg = 1, thus a0b0 = 1, so a0 and b0 are Polynomial rings units. We need to prove that a1;:::; an are nilpotent. Zero-divisors, Since fg = 1, except for the constant coefficient, the coefficients of fg are nilpotents, units Degree zero, in particular Evaluation Coefficients in a anbm = 0 domain an-1bm + anbm-1 = 0 Coefficients in a field an-2bm + an-1bm-1 + anbm-2 = 0 . a0bn + a1bn-1 + ··· + anb0 = 0 Abstract Algebra, Lecture 12 Jan Snellman Proof (cont) Multiply the eqn an-1bm + anbm-1 = 0 Polynomial rings 2 Zero-divisors, by an to conclude that anbm-1 = 0. nilpotents, units Degree Multiply the eqn Evaluation Coefficients in a an-2bm + an-1bm-1 + anbm-2 = 0 domain 2 2 3 Coefficients in a by an to conclude (using anbm-1 = 0) that anbm-2 = 0. field m+1 m+1 Continue until you reach an b0 = 0. Since b0 is a unit, an = 0, so an is nilpotent. n Now f - anx is a unit, so repeat the previous procedure to conclude that an-1 is nilpotent, and so on. Abstract Algebra, Lecture 12 Jan Snellman Lemma If x; y are nilpotent, then so is x + y. Polynomial rings Zero-divisors, nilpotents, units Proof. Degree n n Evaluation Suppose that x = y = 0. Then Coefficients in a 2n domain 2n (x + y)2n = xj y 2n-j ; Coefficients in a j j=0 field X and either j or 2n - j are ≥ n. Abstract Algebra, Lecture 12 Lemma Jan Snellman If x unit, r nilpotent, then x - y (and x + y) is a unit. Proof. Polynomial rings Assume xx-1 = 1, y n = 0. Then Zero-divisors, nilpotents, units n-1 n Degree (1 - y)(1 + y + ··· + y ) = 1 - y = 1; Evaluation Coefficients in a so x-1(x - y) = x-1(1 - yx-1) has inverse domain Coefficients in a x-1(1 + (yx-1) + ··· + (yx-1)n-1); field hence (x - y) has inverse 1 + (yx-1) + ··· + (yx-1)n-1: Abstract Algebra, Lecture 12 Jan Snellman Definition n If f = a0 + a1x + ··· + anx 2 L[x], with an 6= 0, then the degree of f is deg f = max(Supp (f )) = n: Polynomial rings Zero-divisors, nilpotents, units The degree of the zero polynomial is - . Degree We define the Evaluation • leading term as a xn, 1 Coefficients in a n domain • leading monomial as xn, Coefficients in a • field leading coefficient as an Example 4 5 5 If Z[x] = 1 - 3x + 17x then the degree is 5, the l.t. is 17x , the l.c. is 17, and the l.m. is x5. Abstract Algebra, Lecture 12 Theorem Jan Snellman Let f ; g 2 L[x]. If L is a domain then • deg(f + g) ≤ • deg(fg) = deg(f ) + deg(g, max(deg(f ); deg(g)), Polynomial rings • • lt lt lt Zero-divisors, deg(fg) ≤ deg(f ) + deg(g. (fg) = (f ) (g) nilpotents, units Degree Evaluation Example Coefficients in a domain In Z4[x], Coefficients in a field (2x2 + x + 1)2 = 4x4 + x2 + 1 + 4x3 + 4x2 + 2x = x2 + x + 1; so the degree of products can drop in the presence of zero-divisors. In Q[x], (2x2 + x + 1) + (-2x2 - x + 1) = 2; so the degree of sums may drop even with field coefficients. Abstract Algebra, Lecture 12 Jan Snellman Definition n Let f (x) = a0 + a1x + ··· + anx 2 L[x]. Let u 2 L. We define the evaluation of f at u by Polynomial rings n Zero-divisors, f (u) = a0 + a1u + ··· + anu 2 L nilpotents, units Degree Evaluation Coefficients in a Theorem domain For u 2 L, the map Coefficients in a field evu : L[x] L f (x) 7 f (u) ! is a ring homomorphism. ! Abstract Algebra, Lecture 12 Example Jan Snellman A fixed polynomial f (x) 2 L[x] defines a polynomial function f : L L u 7 f (u) Polynomial rings Zero-divisors, ! nilpotents, units That is an importan topic that we will only touch upon in this course. We Degree ! Evaluation note the following oddity: for polynomials in Q[x], different polynomials Coefficients in a give rise to different functions Q Q. However, already for polynomials domain 2 with field coefficients this need not hold. In Z2[x], if f (x) = (x + x)g(x) Coefficients in a for any g(x) 2 [x], then field Z2 ! 2 f ([0]2) = ([0]2 + [0]2)(g([0]2) = [0]2 2 f ([1]2) = ([1]2 + [1]2)(g([1]2) = [0]2 so infinitely many polynomials represent the constantly zero polynomial function. Abstract Algebra, Lecture 12 Jan Snellman Corollary Polynomial rings The set Zero-divisors, nilpotents, units Iu = f f (x) 2 L[x] f (u) = 0 g Degree Evaluation is an ideal in L[x], and Coefficients in a L[x] domain ' L Iu Coefficients in a field The ideals containing Iu are in bijection with the ideals of L. Abstract Algebra, Lecture 12 Jan Snellman Recall: Polynomial rings Theorem Coefficients in a domain n Suppos that L is a domain, and that f = a0 + a1x + ··· + anx 2 L[x]. Divisibility Polynomial rings in 1 L[x] is a domain several variables Coefficients in a 2 f is invertible iff a0 is a unit and deg(f ) = 0.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    38 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us