3*. Let (F,+) be the group of polynomials over3 X {x, a2, a3,...} be the set of all positive integer powers of . Describe the subgroup (X) the field F, under polynomial addition. Let

Q: 30. Let E be an extension field of a finite field F, where F has q elements. Let a e E be algebraic…

A: The objective is to prove that, if  is a finite field and  has  elements. If  be an extension field…

Q: 30. (Remainder Theorem) Let f(x) e F[x] where F is a field, and let a E F. Show that the remainder…

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Q: If F is a field of order n, what is the order of F*?

A: Let F be a finite field  then its order is pn

Q: Let F be a field and α an element not in F such that α^3 - 3α + 2 = 0. Consider the field extension…

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Q: Can you me please Give an example of a ring R and polynomials f (x) and g(x) in R[x] such that…

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Q: Give an example of a ring Zn and polynomials f, g ∈ Zn[x] such that f, g do not satisfy relation…

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Q: M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring…

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Q: 19. Show that if R, R', and R" are rings, and if o : R R' and : R'→ R" are homomorphisms, then the…

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Q: 6. Show that Q(V3) = {a + bv3 : a, b e Q} is a field.

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Q: 10. Show that every irreducible polynomial in Z,[x] is a divisor of xP" 11. Let F be a finite field…

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Q: B. Show that each homomorphism from a field to a ring is either one to one or maps everything nnto 0…

A: 18 Suppose we have a homomorphism φ : F → R where F is a field and R is a ring (for example R itself…

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