Let K be a field estension of a field F and let a1, a2,.....an be elements in K which are algebric over F. Then F(a1, a2.......an) is a finite extension of F and hence an algebric extension of F

Q: Prove if  K and E be the extension of a field F with  [K:F] finite and assume that both K and E are…

A: Let K and E be the extension of a field F with [K:F] finite and assume that both K and E…

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 f: RS be a nontrivial homomorphism from a field R onto a ring S. Prove that S is a field.

A: The given question is related with abstract algebra. Given that f : R → S be a nontrivial…

Q: Theorem 31. Let F be an ordered field with ordered subfield Q. Then F is Archimedean if and only if…

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Q: Theorem 11: Let F/k be a field extension and a F. If α is transcendental over k, then k(a)~k(x).

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Q: 2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K such that f = a²).…

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Q: If F is an ordered field and a ∈ F such that 0 ≤ a 0, then a = 0.

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Q: If R is a U.F.D., then R[x, y] is a U.F.D. If F is a field, then F [x, y] is à U.F.D.

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Q: An ordered field is a field F, and a subset PCF, called a positive set, such that (a) If x, y EP,…

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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: Let F/K be a field extension and a, b in F.  If a has degree m and b ne 0 has degree n over K, prove…

A: The data given in this question are: F/K is a field extension. a and b are elements of F. a has a…

Q: Let K be a finite extension field of a finite field F. Show that there isan element a in K such that…

A: As per the question we are given a field F and its extension K. Then we have to prove that there…

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…