If a , b , m and n are positive integers such that a m and b n are irrational numbers, how can we prove that the sum a m + b n is also irrational?

Question

If   a  ,  b  ,  m and   n are positive integers such that       a    m   and       b    n   are irrational numbers, how can we prove that the sum       a    m    +      b    n   is also irrational?

tektonikafrs · Accepted Answer

Let       b    ′   and       n    ′   be positive integers such that             a      ′              m      ′       and             b      ′              n      ′       are irrational. Let   b and   n be the minimal positive integers such that       a    m    =            a      ′              m      ′      and       b    n    =            b      ′              n      ′      , so that their minimal polynomials are       f    a    =      X    m    −  a and       f    b    =      X    n    −  b, respectively. Note that   m  ,  n  &amp;gt;  1 as       a    m   and       b    n   are irrational.Suppose       a    m    +      b    n    =  q  ∈      Q  . Then       a    m   and       b    n   are roots of       f    b    (  q  −  X  ) and       f    a    (  q  −  X  ), respectively, which shows that       f    a   divides       f    b    (  q  −  X  ) and       f    b   divides       f    a    (  q  −  X  ), respectively. In particular we see that   m  ≤  n and   n  ≤  m, so   n  =  m, and hence       f    a    =  c      f    b    (  q  −  X  ) for some nonzero constant   c  ∈      Q  . Then      X    m    −  a  =      f    a    =  c      f    b    (  q  −  X  )  =  c  (  q  −  X      )    n    −  c  b  =  c  (  q  −  X      )    m    −  c  b  ,which immediately shows that   q  =  0  because   m  ,  n  &amp;gt;  1. It follows that   c  =  ±  1 and   a  =  c  b. Because   a and   b is positive it follows that   c  =  1 and so      a    m    +      b    n    =  2      a    m    =  2      b    n    ,which is irrational because       b    n   is, by assumption.

If a,b,m and n are positive integers such that root(m)(a) and root(n)(b) are irrational numbers, how can we prove that the sum root(m)(a)+root(n)(b) is also irrational?

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