Question

In: Advanced Math

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a)...

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a) Prove that Z[ √ 2] is a subring of R. (b) Find a unit in Z[ √ 2] that is different than 1 or −1.

Solutions

Expert Solution


Related Solutions

Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be...
Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be the relation define on A where (a, b)R(c, d) means that 2 a + d = b + 2 c. a. Prove that R is an equivalence relation. b. Find the equivalence classes [(−1, 1)] and [(−4, −2)].
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2. Prove that if 2p−1 is prime, then p must also be prime. (Abstract Algebra)
Let G = Z x Z and H = {(a, b) in Z x Z |...
Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b} a. Prove directly that H is a normal subgroup in G (use the fact that closed under composition and inverses) b. Prove that G/H is isomorphic to Z8. c. What is the index of [G : H]?
Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A...
Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A countable?
Let A = Z and let a, b ∈ A. Prove if the following binary operations...
Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity). (a) (3 points) f(a, b) = a + b + 1 (b) (3 points) f(a, b) = a(b + 1) (c) (3 points) f(a, b) = x2 + xy + y2
Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if...
Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint:...
let n belongs to N and let a, b belong to Z. prove that a is...
let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.
Discrete Math Course. On Z, let B be the set of subsets A of Z where...
Discrete Math Course. On Z, let B be the set of subsets A of Z where either A is finite or A complement is finite. Define + and * as union and interception. Show whether or not B is a boolean algebra.
Let ∼ be the relation on P(Z) defined by A ∼ B if and only if...
Let ∼ be the relation on P(Z) defined by A ∼ B if and only if there is a bijection f : A → B. (a) Prove or disprove: ∼ is reflexive. (b) Prove or disprove: ∼ is irreflexive. (c) Prove or disprove: ∼ is symmetric. (d) Prove or disprove: ∼ is antisymmetric. (e) Prove or disprove: ∼ is transitive. (f) Is ∼ an equivalence relation? A partial order?
Problem 3.Let Σ be the top hemisphere (z≥0) of x^2+y^2+z^2= 9 with upward normal and let...
Problem 3.Let Σ be the top hemisphere (z≥0) of x^2+y^2+z^2= 9 with upward normal and let F(x,y,z) =xzi+yj−k. Evaluate ∫∫ΣF·dS directly.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT