Suppose (Z, +, ·) is an ordered integral domain. Let a and b and c be...
Suppose (Z, +, ·) is an ordered integral domain. Let a and b and c be elements of Z such that c^2 + a · c + b = 0.
(a) Prove that if 4b = a^2 , then x^2 + a · x + b ≥ 0 for all x ∈ Z.
(b) Prove that if 4b =/= a^2 , then there is exactly one element d in Z such that d^2 + a · d + b = 0 and d =/= c.
(c) For which x ∈ Z is x^2 + a · x + b > 0? As usual, provide a proof
Homework Answers
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Suppose (Z, +, ·) is an ordered integral domain. Let a and b and
c be...
8. Let A be an integral domain containing elements x, y, and z. Prove the following...
8. Let A be an integral domain containing elements x, y, and z. Prove the following facts. (a) If z|x and zly, then x/2 + y/2 = (x + y)/2. (b) If 2 x, then y. (x/2) = (y • x)/2. (c) If yız and x[(z/y), then (x • Y)|z, and 2/(x • y) = (z/y)/x.
1,(Z) = { a bla, b, c, d. Let M2(Z) = a, b, c, d e...
1,(Z) = { a bla, b, c, d. Let M2(Z) = a, b, c, d e Z} with matrix addition and multiplication. Which of the following is true: it is a commutative ring with unity it is a ring with unity but not commutative it is a ring without unity and not commutative it is a commutative ring without unity Question 4 Which of the following statement is true about the ring of integers (with usual addition and multiplication) the...
ifD is an ordered integral domain with positive elements D^p and unity e. prove if a∈D then a>a-e
ifD is an ordered integral domain with positive elements D^p and unity e. prove if a∈D then a>a-e
2. Let R be an integral domain containing a field K as a unital subring. (a)...
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain...
Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain and R* = {+1}. (b) Show that irreducibles of Rare Ep for primes pe Z, and S() ER with (0 €{+1} which are irreducible in Q[r]. (c) Show that r is not a product of irreducibles, and hence R does not satisfy the ascending chain condition for principal ideals.
37. Show that if D is an integral domain, then 0 is the only nilpotent element...
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
Prove Cauchy's Integral Theorem for k-connected Jordan domains: Let I be a k-connected Jordan domain and...
Prove Cauchy's Integral Theorem for k-connected Jordan domains: Let I be a k-connected Jordan domain and f(2) be analytic in some domain containing 12. Then, Son f(z)dz = 0. Hint: Use the Deformation Principle.
3. Let a, b, c E Z such that ca and (a,b) = 1. Show that...
3. Let a, b, c E Z such that ca and (a,b) = 1. Show that (c, b) = 1. 4. Suppose a, b, c, d, e E Z such that e (a - b) and e| (c,d). Show that e (ad — bc). 5. Fix a, b E Z. Consider the statements P: (a, b) = 1, and Q: there exists x, y E Z so that ax + by = 1. Bézout’s lemma states that: if P, then...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by ...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is uni...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
8. Let A be an integral domain containing elements x, y, and z. Prove the following facts. (a) If z|x and zly, then x/2 + y/2 = (x + y)/2. (b) If 2 x, then y. (x/2) = (y • x)/2. (c) If yız and x[(z/y), then (x • Y)|z, and 2/(x • y) = (z/y)/x.
1,(Z) = { a bla, b, c, d. Let M2(Z) = a, b, c, d e Z} with matrix addition and multiplication. Which of the following is true: it is a commutative ring with unity it is a ring with unity but not commutative it is a ring without unity and not commutative it is a commutative ring without unity Question 4 Which of the following statement is true about the ring of integers (with usual addition and multiplication) the...
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
Let R={1 € Q[2] : [0) € Z}. (a) Show that R is an integral domain and R* = {+1}. (b) Show that irreducibles of Rare Ep for primes pe Z, and S() ER with (0 €{+1} which are irreducible in Q[r]. (c) Show that r is not a product of irreducibles, and hence R does not satisfy the ascending chain condition for principal ideals.
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
Prove Cauchy's Integral Theorem for k-connected Jordan domains: Let I be a k-connected Jordan domain and f(2) be analytic in some domain containing 12. Then, Son f(z)dz = 0. Hint: Use the Deformation Principle.
3. Let a, b, c E Z such that ca and (a,b) = 1. Show that (c, b) = 1. 4. Suppose a, b, c, d, e E Z such that e (a - b) and e| (c,d). Show that e (ad — bc). 5. Fix a, b E Z. Consider the statements P: (a, b) = 1, and Q: there exists x, y E Z so that ax + by = 1. Bézout’s lemma states that: if P, then...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
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