10. Prove that the ring Z[w] is a UFD, where w²+w+1=0. [Hint: As in the previous question, prove that Z[w] has a division algorithm with respect to the function "absolute value squared" v(a + bw) = (a + bw) (a + bw²) = a² - ab+b².]

Q: Use the norm map N: Z [√-3] → Z2o by N(a+b√-3) = a² + 36² (you can assume it is multiplicative, i.e…

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Q: Let p be a prime. Are there any nonconstant polynomials in Zp[x]that have multiplicative inverses?…

A: Given: A prime number p. To check: Whether any nonconstant polynomials in Zp[x] have multiplicative…

Q: 3. In the ring F2[x], consider the polynomials f(x) = x³ +x² + 1, g(x) = x²+1 (a) Run the extended…

A: A) Therefore, the GCD ( d(x) = x + 1 ), and the Bézout identity holds with ( a(x) = x ) and ( b(x) =…

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: 4. Show that 7 is irreducible in the ring Z[V5] using the norm N defined by N(a + bv5) = | a? –…

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Q: any cubic polynomial f = K[r] can be written in the form f = r³+ar+ b. change of varillible

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Q: Let a be the positive real fourth root of 3. Factor the polynomial x¹-3 into irreducible factors in…

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Q: 4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of…

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Q: don't understand how the equation turned into 5x+x+6/x(x+6), could you go through the equation again…

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Q: (d) Show that the division algorithm holds in Z[V2]. Hint: Let a, 3 e Z[v2] with B+ 0. Write a/3 =r+…

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Q: 1. Find irr(a, Q) and deg(a, Q) for the given algebraic number a = √2+ i C. [Hint: Find irr(a, Q) as…

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Q: 11. Determine if the following polynomials are reducible or irreducible over Z7: a) x³ + 5x² + 4x +…

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Q: Show that, for all constructible numbers y € R, x³ – 2 is irreducible over • Q(r).

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Q: 1. Factor each of the following polynomials completely into irreducible polynomials: a) x² + 1 in…

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Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Prove that Z[√ 5] is not a unique factorization domain.

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Q: 5. Explain why each of the polynomials below is irreducible in the indicated polynomial ring. Make…

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Q: Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R[x] by defining an+1,&n+2,...…

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Q: 3. Let R be the subring (a+b√10 | a,b e Z) of the field of real numbers. (a) The map N:R-Z given by…

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Q: The ring Z[/2] is an ED with d(a + bv2) = |a² – 21³| . Show that the equations r² – 2y² = 1 and r² –…

A: The objective is to show that equations x2-2y2=1  and  x2-2y2=-1  each have infinitely many integer…

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

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Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: Let Q be the field of rational numbers and let F √2, √7, √14 are irrational. = Q(√2, √7). You may…

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Q: 2. Let meN and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and…

A: Let m in N and a in Z.(a) If gcd(a, m) = 1, then Bézout's lemma gives the existence of integers x…

Q: Let a E Z, be a zero of the irreducible polynomial x + x +1 over Z,. Show that the Frobenius…

A: The given problem is related to abstract algebra. Given: α∈ ℤ2 is a zero of the irreducible…

Q: Let z be a root of z5 – 1 = 0 with z # 1. Compute %3D 215 + 216 + z17 +· . + z50

A: Since z is a root of z5-1=0 ⇒z-11+z+z2+z3+z4=0⇒1+z+z2+z3+z4=0…

Q: how that both √2 and i are in Q(i + √2) and conclude [Q(i + √2) : Q] = 4. Find the minimal olynomial…

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Q: Prove that in a Euclidean ring R, (a, b) can be found as follows : b = q0 a + r¡, where d (r¡)<d (a)…

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Q: Find the inverse of 1 + √2 in Z[√2]. What is the multiplicativeorder of 1 + √2?

A: Here ℤ2=a+b2/a, b∈ℤ We need to find multiplicative inverse of 1+2 in  ℤ2. An element b∈R is said to…

Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

Q: Let g(x) be a polynomial in Z₂[x]. Prove that if the polynomial code C generated by g(x) with length…

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Q: Determine if the following polynomials are- irreducible in

A: In the given question, the concept of Irreducible Polynomial is applied. Irreducible Polynomial An…

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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