Polynomial Rings Yohana Solomon November 15, 2016 Yohana Solomon Polynomial Rings November 15, 2016 1 / 35 Outline 1 Definitions and Basic Properties 2 Polynomial Rings over Fields 3 Polynomial Rings that are Unique Factorization Domains 4 Irreducibility Criteria Yohana Solomon Polynomial Rings November 15, 2016 2 / 35 Definitions and Basic Properties Yohana Solomon Polynomial Rings November 15, 2016 3 / 35 Multiplication is performed by first defining (ax i )(bx j ) = abx i+j and then extending to all polynomials by the distributive laws so that in general n ! m ! n+m k ! X i X i X X k ai x × bi x = ai bk−i x : i=0 i=0 k=0 i=0 In this way R[x] is a commutative ring with identity (the identity 1 from R) in which we identify R with the subring of constant polynomials. Addition of polynomials is componentwise: n n n X i X i X i ai x + bi x = (ai + bi )x : i=0 i=0 i=0 Basic Properties The polynomial ring R[x] in the indeterminate x with coefficients from R is the set of all formal sums n n−1 anx + an−1x + ··· + a1x + a0 with n ≥ 0 and each ai 2 R. Yohana Solomon Polynomial Rings November 15, 2016 4 / 35 Multiplication is performed by first defining (ax i )(bx j ) = abx i+j and then extending to all polynomials by the distributive laws so that in general n ! m ! n+m k ! X i X i X X k ai x × bi x = ai bk−i x : i=0 i=0 k=0 i=0 In this way R[x] is a commutative ring with identity (the identity 1 from R) in which we identify R with the subring of constant polynomials. Basic Properties The polynomial ring R[x] in the indeterminate x with coefficients from R is the set of all formal sums n n−1 anx + an−1x + ··· + a1x + a0 with n ≥ 0 and each ai 2 R. Addition of polynomials is componentwise: n n n X i X i X i ai x + bi x = (ai + bi )x : i=0 i=0 i=0 Yohana Solomon Polynomial Rings November 15, 2016 4 / 35 In this way R[x] is a commutative ring with identity (the identity 1 from R) in which we identify R with the subring of constant polynomials. Basic Properties The polynomial ring R[x] in the indeterminate x with coefficients from R is the set of all formal sums n n−1 anx + an−1x + ··· + a1x + a0 with n ≥ 0 and each ai 2 R. Addition of polynomials is componentwise: n n n X i X i X i ai x + bi x = (ai + bi )x : i=0 i=0 i=0 Multiplication is performed by first defining (ax i )(bx j ) = abx i+j and then extending to all polynomials by the distributive laws so that in general n ! m ! n+m k ! X i X i X X k ai x × bi x = ai bk−i x : i=0 i=0 k=0 i=0 Yohana Solomon Polynomial Rings November 15, 2016 4 / 35 Basic Properties The polynomial ring R[x] in the indeterminate x with coefficients from R is the set of all formal sums n n−1 anx + an−1x + ··· + a1x + a0 with n ≥ 0 and each ai 2 R. Addition of polynomials is componentwise: n n n X i X i X i ai x + bi x = (ai + bi )x : i=0 i=0 i=0 Multiplication is performed by first defining (ax i )(bx j ) = abx i+j and then extending to all polynomials by the distributive laws so that in general n ! m ! n+m k ! X i X i X X k ai x × bi x = ai bk−i x : i=0 i=0 k=0 i=0 In this way R[x] is a commutative ring with identity (the identity 1 from R) in which we identify R with the subring of constant polynomials. Yohana Solomon Polynomial Rings November 15, 2016 4 / 35 the units of R[x] are just the units of R R[x] is an integral domain. Proposition: Let R be an integral domain. Then degree p(x)q(x) = degree p(x) + degree q(x) if p(x), q(x) are nonzero Yohana Solomon Polynomial Rings November 15, 2016 5 / 35 R[x] is an integral domain. Proposition: Let R be an integral domain. Then degree p(x)q(x) = degree p(x) + degree q(x) if p(x), q(x) are nonzero the units of R[x] are just the units of R Yohana Solomon Polynomial Rings November 15, 2016 5 / 35 Proposition: Let R be an integral domain. Then degree p(x)q(x) = degree p(x) + degree q(x) if p(x), q(x) are nonzero the units of R[x] are just the units of R R[x] is an integral domain. Yohana Solomon Polynomial Rings November 15, 2016 5 / 35 Definition: The polynomial ring in the variables x1; x2; :::; xn with coefficients in R, denoted R[x1; x2; :::; xn] = R[x1; x2; :::; xn−1][xn] Yohana Solomon Polynomial Rings November 15, 2016 6 / 35 Example: Let p(x; y; z) = 2x 2y − 3xy 3z + 4y 2z 5 and q(x; y; z) = 7x 2 + 5x 2y 3z 4 − 3x 2z 3 be polynomials in Z[x; y; z]. Note: The polynomial ring Z[x; y; z] in three variables x, y and z with integers coefficients consists of all finite sums of monomial terms of the form ax i y j z k (of degree i + j + k). sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Yohana Solomon Polynomial Rings November 15, 2016 7 / 35 sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Write each of p and q as a polynomial in x with coefficients in Z[y; z]. sage: R2 = QQ['y,z']['x'] sage: R2(px) (2 ∗ y) ∗ x 2 − (3 ∗ y 3 ∗ z) ∗ x + 4 ∗ y 2 ∗ z 5 sage: R2(qx) (5 ∗ y 3 ∗ z 4 − 3 ∗ z 3 + 7) ∗ x 2 Yohana Solomon Polynomial Rings November 15, 2016 8 / 35 sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Find the degree of p and q. sage: px.degree() 7 sage: qx.degree() 9 Yohana Solomon Polynomial Rings November 15, 2016 9 / 35 sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Find the degree of p and q in each of the three variables x, y and z. sage: px.exponents() [(0, 2, 5), (1, 3, 1), (2, 1, 0)] sage: qx.exponents() [(2, 3, 4), (2, 0, 3), (2, 0, 0)] Yohana Solomon Polynomial Rings November 15, 2016 10 / 35 sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Compute pq and find the degree of pq in each of the three variables x, y and z. sage: rx = px*qx; rx 20∗x 2 ∗y 5 ∗z 9 −15∗x 3 ∗y 6 ∗z 5 +10∗x 4 ∗y 4 ∗z 4 −12∗x 2 ∗y 2 ∗ z 8+9∗x 3∗y 3∗z 4+28∗x 2∗y 2∗z 5−6∗x 4∗y ∗z 3−21∗x 3∗y 3∗z+14∗x 4∗y sage: rx.degree() 16 sage: rx.exponents() [(2, 5, 9), (3, 6, 5), (4, 4, 4), (2, 2, 8), (3, 3, 4), (2, 2, 5), (4, 1, 3), (3, 3, 1), (4, 1, 0)] Yohana Solomon Polynomial Rings November 15, 2016 11 / 35 sage: R1 = QQ['x,y,z'] sage: (x,y,z) = R1.gens() sage: px = 2 ∗ x 2 ∗ y − 3 ∗ x ∗ y 3 ∗ z + 4 ∗ y 2 ∗ z 5; sage: qx = 7 ∗ x 2 + 5 ∗ x 2 ∗ y 3 ∗ z 4 − 3 ∗ x 2 ∗ z 3; Write pq as a polynomial in the variable z with coefficients in Z[x; y]. sage: R3 = QQ['x,y']['z'] sage: R3(rx) 20∗x 2 ∗y 5 ∗z 9 −12∗x 2 ∗y 2 ∗z 8 +(−15∗x 3 ∗y 6 +28∗x 2 ∗y 2)∗ z 5+(10∗x 4∗y 4+9∗x 3∗y 3)∗z 4−6∗x 4∗y ∗z 3−21∗x 3∗y 3∗z+14∗x 4∗y Yohana Solomon Polynomial Rings November 15, 2016 12 / 35 Polynomial Rings over Fields Yohana Solomon Polynomial Rings November 15, 2016 13 / 35 Theorem: Division Algorithm Let F be a field. The polynomial F [x] is a Euclidean Domain.