Un curso de álgebra. Gabriel Navarro Ortega

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entero, entonces 4 no divide a n² + 2.

      25. Si a, b, c son enteros no cero y mcd(a, c) = 1, probar que mcd(a, b) = mcd(a, bc).

      26. Recordar que si a ∈ ℝ − ℚ, entonces a se dice irracional.

      (i) Sean a ∈ ℚ y b ∈ ℝ irracional. Probar que a + b es irracional. Si a ≠ 0, probar que ab es irracional.

      (ii) Si n ∈ ℕ, probar que es irracional.

      (iii) Probar que es irracional.

      (iv) Probar que no se puede escribir de la forma , donde r, s ∈ ℚ.

      27. Comprobar que existen números irracionales a, b ∈ ℝ tales que a^b es racional.

      (Ayuda: Si no es racional, volver a elevar a ).

      28. Si z = a + bi, entonces el conjugado complejo de z es = a − bi. El módulo de z es , probar lo siguiente:

      (i) z₁z₂ = z₂z₁.

      (ii) z₁(z₂z₃) = (z₁z₂)z₃.

      (iii) z₁(z₂ + z₃) = z₁z₂ + z₁z₃.

      (iv)

      (v)

      (vi)

      (vii)

      (viii)

      (ix)

      29. Hallar las ráıces 8-ésimas de la unidad.

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