Let K be an extension of a field F. If an) is a finite an e K are algebraic over F, then F (a1, a2, . ... tension of F and so is algebraic over F.

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: Let K be a field extension field F and let a € K be algebric over F.

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Q: Abstract Algebra. Please explain everything in detail.

A: To prove the stated properties pertaining to the subfields of the given field

Q: Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple…

A: Let, α∈E be such that α∉F. As we know that, If E is the finite extension field F and K is finite…

Q: Let E be a finite extension of R. Use the fact that C is algebraicallyclosed to prove that E = C or…

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Q: Let K be an extension of a field F. An element a e K is algebraic over F if and only if [F (a) : F]…

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Q: Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.

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Q: Let K be an extension of a field F. An element a e K is algebraic over F if and only if [F (a): F]…

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Q: Let F be a field and a be a non-zero element in F. If f(x) is reducible over F, then f(x+a)EF[x] is…

A: Use the properties of ring of polynomials to solve this problem.

Q: If F is a finite field of characteristic p, then aap is an automorphism of F.

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Q: Show that if E is an algebraic extension of a field F and contains all zeros in F of every f(x) E…

A: If E is an algebraic extension of a field F and contains all zeros in F¯  of every fx∈Fx, then E is…

Q: Define an algebraically closed field. Show that field E is algebraically closed if and only if every…

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Q: Let A,B ∈Mn×n(F) be such that AB= −BA. Prove that if n is odd and F is not a field of characteristic…

A: The matrix is not invertible if its determinant is 0.The determinant of the product of the matrices…

Q: et FCK be a field extension and let R be the alg K. Then R is a subfield of K and FCR.

A: let alpha and beta are in R

Q: Let F be a field and let a be a nonzero element of F. (a) If af(x) is irreducible over F, prove that…

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Q: If 0±x#1 in a field R, then x is an idempotent. но чо

A: Only idempotent element in a field are 0 and 1

Q: Find an example of a field F and elements a and b from someextension field such that F(a, b) ≠ F(a),…

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Q: If u is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(EU F)+µ(EnF) %3D

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Q: If K be an extension of a field F then the mapping : F[x]-f(a) defined by (h(x)) = h(a) is Both (A)…

A: It is given that K be an extension of a field F and  ϕ : Fx→fa defined by  ϕhx = ha . We have to…

Q: A field that has no proper algebraic extension is called algebraically closed. In 1799, Gauss proved…

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Q: Let K be a splitting field of f(x) = x* + 2x² 6. over Q find the Galois group Gal(*/o). and compute…

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Q: Let (S, +,) be a subfield of the field (F, +,), then (S, +,) is a) integral domain b) field c)…

A: Hello, learner we can answer first question as per the honor policy. Please resubmit other question…

Q: et K be an extension of a field F. If a and b in K are alg ma ±b, ab and (b+0) are algebric over F…

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Q: Let ø : F → R be a ring homomorphism where F is a field. Show that either o is one-to-one or ø is…

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Q: 3. Suppose E is a splitting field of g(x) E F[x] \ F over F. Suppose E = F[a] for some a. Prove that…

A: Let σ∈EmbFE,E.  Given that E=Fα. That is α is algebraic over F and mα,Fx=a0+a1x+...+akxk is the…

Q: Label each of the following statements as either true or false.   Let F be a field. If p(x) is…

A: Given that, the statement Let F be a field. If p(x) is reducible over F, the quotient ring F [x…

Q: Let p be a prime, F = Zp(t) (the field of quotients of the ring Zp[x])and f(x) = xp - t. Prove that…

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

A: We have to prove that F(ꭤ) has qn elements.

Q: If K is a finite field extension of a field F and L is a finite field extension of K. then L is a…

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Q: Show that if E is an algebraic extension of a field F and contains all zeros in \bar{F} of every  f…

A: To show:

Q: Let K be a field estension of a field F and let a1, a2,.....an be elements in K which are algebric…

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Q: Suppose that K is an extension field of F with a, B,y E K Prove that if {a, B, 7} is linearly…

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Q: Suppose E is an extension of a field F, and a,b are elements of E. Further, assume a is algebraic…

A: suppose E is an exiension of a field F , a,b are element of E. assume  a is algebraic over F of…

Q: Let F be an ordered field with the least upper bound property. Prove that there is a unique function…

A: An ordered field F with least upper bound property To prove that exist a unique function s.t. (i)…

Q: Let F be a field with char (F) = p > 0. Prove that F is perfect if and only if the homomorphism %3D…

A: Suppose that F is a perfect field. We want to show that each α∈F has a pth root in F. Since F is…

Q: Let F be a field. Prove that Fl) E F.

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Q: Let p be a prime and consider the finite field F = F,2 and its extension E = F,20. What is…

A: When F is a subfield of a field F, E/F is the field extension. In that case, E is considered as the…

Q: Let F be a field. Prove that for every integer n > 2, there exist r, sE F such that x² + x + 1 is a…

A: Given the statement Let F be a field. We have to Prove that for every integer n >= 2 , there…

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Let degree of the polynomial fx is n and degree of the polynomial gx is m. Given that fx, gx are…

Q: Prove that every ideal in F[x], where F is a field, is a principal ideal

A: To show: Every ideal in F[x], where F is a field is a principle ideal

Q: Let F be a field and let p(x), a1(x), a2(x), . . . , ak(x) ∈ F[x], wherep(x) is irreducible over F.…

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Q: Show that if [E:F]=2, then E is a splitting field over F.

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Q: Let F be a field and let a be a nonzero element of F.a. If af(x) is irreducible over F, prove that…

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Q: Suppose that F < K < E, then E is a splitting field over F. Prove that K is not a splitting field…

A: Given: F≤K≤E E is a splitting field over F   To prove: K is not a splitting field over F

Q: Show that Let K be an extension of a field F and a e K be algebraic over F. Then F[a] = F (a), where…

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Q: If µ is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(Eu F)+µ(En F) %3D

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Q: Let E be a field and , 6E E be nonzero polynomials. (a) If ab and a, prove that a = db for some…

A: Let E be a field and a, b ∈ E[x] be non-zero polynomials.

Q: If F is a field, then it has no proper ideal. T OF

A: I have given the answer in the next step. Hope you understand that

Q: If p(x)∈F[x] and deg p(x) = n, show that the splitting field for p(x)over F has degree at most n!.

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Q: Let ϕ : F → R be a ring homomorphism from a field F into a ring R. Prove that if ϕ ( a ) = 0  for…

A: Consider ϕ : F → R be a ring homomorphism from a field F into a ring R,Since, ϕ ( a ) = 0  for some…