Without using the Primitive Element Theorem, prove that if [K:F]is prime, then K has a primitive element.

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A: Part (A):- Given information 1.  fx =7-x2.  7<343x-23.  fx =x2+1

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Q: (b) (c) (d) (e) dy=3x²(y-1) dx dy_y+2 = dx x+1 dy dx -+e*y² = 0 dy dx -= cos x tan y hotnes

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Q: Suppose that a group of order 8 has exactly five elements of order 2.Identify the group.

A: Abstract Algebra, Dihedral group

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Q: Calculus    Verify Clairauts theorem for f(x,y) = 2xe^y-3ye^-x

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Q: Examples. Justify your answers briefly. (c) Give an example of series that ΙΣ n=0 n=0 an and an +bn…

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Q: Assume that the following row operations R₁ + R₂ R₂ +2R₁ R₂ R₁+R₂ R₁ +5R₂ → R₂ are used in an…

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Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

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A: Given : L1 , 2T=-2 , 3T  , L1 , -1T=5 , 2T To Find : L7, 5T

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Q: Suppose that a is algebraic over a field F. Show that a and 1 + a-1have the same degree over F.

A: Let's consider that 1+a-1∈Qa then, it can be concluded that, Q1+a-1⊆Qa                  …1

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Q: Are there any nonconstant polynomials in Z[x] that have multiplicativeinverses? Explain your answer.

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Q: M In³ + (n + 1)³ > (n + 2)³ GEGECS

A: Let p(n) be the statement 'n3+(n+1)3>(n+2)3'. 13+23=9 ≯27=33…