5.10. Let K/F be a finite extension of fields. Prove that there is a finite set of elements a₁,..., an E K such that K= F(a₁,..., An). (Hint. Adjoin elements one at a time and estimate the degree of the extension that you get.)

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Q: =Fnxn. Let C be a fixed element of by Tc(A) = AC-CA.

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Q: .3. Let K be an extension of a field F. Let

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Q: 6. Is the function h: Z → Z defined by ( 2n if n 20 h(n) -n if n < 0 one-to-one? Is it onto?

A: The solution is given as follows

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Q: Let f:Z → Z be given by f (x) = 3x + 1. a. Show that f is one-to-one. 7. b. Is f onto? EXPLAIN.

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Q: Prove that the function f from Nx N to N defined by f(m, n) = 2m3 is one-to-one but not onto.

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