Use the field norm to show: a) 1 + √2 is a unit in Z[√2] b) −1+√√-3 -1+√√-3 2 is a unit in Z [ 1+ √−3 ] 2

Q: Find the inverse of X + (X² – 8X + 7) in the form aX + b in the ring Q[X]/(X² – 8X + 7). - a b =…

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Q: Find Tay br Polynomial. for the function fox): Vx + 3. ot degree 3 centered ata=2

A: According to the problem, we have

Q: Show by any means that Q[r]/(x4 + 3x2 + 9x + 1) is a field.

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Q: 2. Find all of the prime quadratic polynomials in F4[x], where F4 {0,1, a, b} be the field of four…

A: Given: F4={0,1,a,b}

Q: Find all the units of: a. R b.Z c. ZP

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Q: • Suppore (12₁+₁ •) is a commurative ring Define an operation on R s.t a ob = a +b-ab associative ?…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (6) The ring (Z,12, +12,·12 ) is ...... (a) field (b) semi-simple (c) simple (d) No Choice O (a) O…

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Q: CE Z3 A) Find all the values that make the Z[x]/(x² +x + c) quotient ring an object. B) Show that…

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Q: What is the order of the splitting field ofx5 + x4 + 1 = (x2 + x + 1) . (x3 + x + 1) over Z2?

A: We have to find the order of the splitting field of x5+x4+1=x2+x+1⋅x3+x+1 over Z2. Let fx=x3+x+1 and…

Q: Find all solutions of x2 + x = 0 in the ring Z12

A: Given x2+x=0

Q: Appendix C: Fields Problem 1. Consider the following set of complex numbers Q(i) = (a + bi: a, b €…

A: The given set is Qi=a+bi : a,b∈ℚ. To Prove: Qi forms a field under addition and multiplication. As…

Q: a. Show that the field Q(v2, V3) = {a + bv2 + cv3 + dvZV3: a, b, c, d E Q} is a finite extension of…

A: a. The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational…

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

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Q: For which fields F listed below is the polynomial X4 + X3 +1 e F[X] irreducible? Select all that…

A: please comment if you need any clarification. if you find my answer useful put thumbs up. Thank you.

Q: Explain why the polynomial rings R[x] and C[x] are not isomorphic.

A: Suppose there exists an isomorphism φ:R[x]→C[x]. Because isomorphisms are by definition surjective,…

Q: Find an a such that (√2, √2) = Q(a). Show that your a is indeed in the given field. Verify by direct…

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Q: 10. When dividing x* + 3x? + 2x + 1 by x² + 2x + 3 in Z5[x], the remainder is а) 0 b) 2 c) 3 d) 4 a)…

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Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: Show that the 8th cyclotomic field is Q(i, √2).

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Q: Use Eisenstein's criterion to show that the polynomial x3 –- 3x? + 9x – 10 is irreducible over the…

A: Solution :-

Q: Check whether i e Q(v-2) or not.

A: Let p,q ∈Q such thatp2-2q2+i22pq=p+q-22=-1then either p=0 or q=0, as22pq=0 Since -1 has no imaginary…

Q: For which fields F listed below is the polynomial X + X3 +1 € F[X] irreducible? Select all that…

A: Given F is a field and fX=X4+X3+1 ∈FX. We have to select the correct option for which the given…

Q: 11 Determine are whether the following polynomials the adjacent fields. irreducible p(x) = on a)…

A: Polynomial a is reducible.