Suppose that a; b; c 2 R with a 6= 0 and b2 ?? 4ac < 0, so
that
r(x) = ax2 + bx + c
is an irreducible quadratic polynomial. Prove that
R[x]=r(x)R[x] =
C :
[Hint: use the Fundamental Homomorphism Theorem. You may assume
with-
out proof that an appropriate evaluation map is a ring
homomorphism.]
![Suppose that a,b,cE R with a?0 and b2-4ac ? 0, so that ba c is an irreducible quadratic polynomial. Prove that Hint: use the Fundamental Homomorphism Theorem. You may assume with- out proof that an appropriate evaluation map is a ring homomorphism.]](http://img.homeworklib.com/questions/c8585b00-839b-11eb-83ce-dddf34a6b636.png?x-oss-process=image/resize,w_560)
Homework Answers
The two roots of the quadratic equation ax2 + bx + c = 0 can be...
The two roots of the quadratic equation ax2 + bx + c = 0 can be found using the quadratic formula as -b+ v62 – 4ac . -b-v6² – 4ac 1 X1 = - and x2 = 2a 2a When b2 – 4ac < 0 this yields two complex roots - -b V4ac – 62 -b Vac – 6² x1 = = +. . 2a 2a i. and x2 = . za 2al Using the quadratic formula the roots of...
Math toolbox: -b+ b2 - 4ac ax2+bx+c =0 solve x= 2a 1 cm3 = 0.001 dm3...
Math toolbox: -b+ b2 - 4ac ax2+bx+c =0 solve x= 2a 1 cm3 = 0.001 dm3 The equilibrium constant is calculated Keq = 0.36 for the reaction @ 400 K when using M (mol/dm3) as the unit for the gasses. note: not the pressure unit. PCL5 (g) <=> PC13 (g) + Cl2 (g) When using M as the units, the Keq is unitless with unit canceled to C@= 1 M. Giving that 4.733 g of PC15 (Mw = 208 g/mol)...
6. Consider the function Q(z) = Az[+ B2:1x2+ Cr where A, B, and C are numbers not all zero, = 0 a...
6. Consider the function Q(z) = Az[+ B2:1x2+ Cr where A, B, and C are numbers not all zero, = 0 and the level sets of the associated quadratic form; and recall the well-known classification rule for conic sections by the discriminant (B2 -4AC): if B2 - 4AC < 0, then the conic is an ellipse; if B2-4AC = 0, then the conic is an parabola; and if B2-4AC > 0, then the conic is a hyperbola. (a) Complete the...
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) =...
Please solve all questions
1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
Please help! Thank you so much!!! 1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-mo...
Please help! Thank you so much!!!
1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
ei0 : 0 E R} be the group of all complex numbers on the unit circle...
ei0 : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let ø : R -> U 1. (30) Let R be the group of real numbers under addition, and let U be the map given by e2Tir (r) (i) Prove that d is a homomorphism of groups (ii) Find the kernel of ø. (Don't just write down the definition. You need to describe explicit subset of R.) an real number r for...
for a matrix solution of the quadratic (3) Find a formula of the form x = -B C equation ax2 + bx +c = 0. Here c denotes...
for a matrix solution of the quadratic (3) Find a formula of the form x = -B C equation ax2 + bx +c = 0. Here c denotes and 0 denotes 0 0 (Hint: First show how the square root of any number D can be obtained using a where it looks different depending matrix of the form on whether D is negative. Then use the quadratic formula.) positive or
for a matrix solution of the quadratic (3) Find a...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...
The two roots of the quadratic equation ax2 + bx + c = 0 can be found using the quadratic formula as -b+ v62 – 4ac . -b-v6² – 4ac 1 X1 = - and x2 = 2a 2a When b2 – 4ac < 0 this yields two complex roots - -b V4ac – 62 -b Vac – 6² x1 = = +. . 2a 2a i. and x2 = . za 2al Using the quadratic formula the roots of...
Math toolbox: -b+ b2 - 4ac ax2+bx+c =0 solve x= 2a 1 cm3 = 0.001 dm3 The equilibrium constant is calculated Keq = 0.36 for the reaction @ 400 K when using M (mol/dm3) as the unit for the gasses. note: not the pressure unit. PCL5 (g) <=> PC13 (g) + Cl2 (g) When using M as the units, the Keq is unitless with unit canceled to C@= 1 M. Giving that 4.733 g of PC15 (Mw = 208 g/mol)...
6. Consider the function Q(z) = Az[+ B2:1x2+ Cr where A, B, and C are numbers not all zero, = 0 and the level sets of the associated quadratic form; and recall the well-known classification rule for conic sections by the discriminant (B2 -4AC): if B2 - 4AC < 0, then the conic is an ellipse; if B2-4AC = 0, then the conic is an parabola; and if B2-4AC > 0, then the conic is a hyperbola. (a) Complete the...
Please solve all questions
1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
Please help! Thank you so much!!!
1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
ei0 : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let ø : R -> U 1. (30) Let R be the group of real numbers under addition, and let U be the map given by e2Tir (r) (i) Prove that d is a homomorphism of groups (ii) Find the kernel of ø. (Don't just write down the definition. You need to describe explicit subset of R.) an real number r for...
for a matrix solution of the quadratic (3) Find a formula of the form x = -B C equation ax2 + bx +c = 0. Here c denotes and 0 denotes 0 0 (Hint: First show how the square root of any number D can be obtained using a where it looks different depending matrix of the form on whether D is negative. Then use the quadratic formula.) positive or
for a matrix solution of the quadratic (3) Find a...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...
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