Another proof that nth roots of integers are irrational, except for perfect nth powers.
(Based on the Rational Root Theorem)
Scott E. Brodie
We begin by recalling two helpful propositions from Euclid.
If two numbers are relatively prime to any number, then their product is also relatively prime to the same.
Proof: Suppose a is relatively prime to both b and c. Since a and b are relatively prime, there exist integers (perhaps negative) m and n such that
Multiplying these two equations together,
|(ma + nb)(ja + kc)||= 1|
|= maja + makc +nbja + nbkc|
|= (maj + mkc + nbj)a + (nk)bc|
so a and bc are relatively prime.
Repeating the argument verifies that if a is relatively prime to b, then a is relatively prime to bn for any positive integer n.
If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original factors.
It is no more work to prove a simple generalization: If integers a and b are relatively prime, and a divides the product bc, then a divides c. [See the aforementioned "Remark" for the proof].
With these tools in hand, we can now prove the Rational Root Theorem, from which the general result on the irrationality of n-th roots follows as a simple corollary:
Rational Root Theorem
Let P(x) be a polynomial with integer coefficients, say
P(x) = anxn + an-1xn-1 + ... + a1x + a0
and suppose that r = c/d is a rational root of P [that is,
Proof: Inserting the argument x = c/d into the expression for P(x) yields
0 = an cn/dn + an-1 cn-1/dn-1 + ... + a1 c/d + a0
Multiplying through by dn and isolating the first term yields
-ancn = an-1cn-1d + ... + a1cdn-1 + a0dn
Since d is a factor of every term on the right hand side of this equation, d must divide ancn. But c and d are relatively prime, so d and cn are relatively prime, and it follows from the generalized version of VII.30 that d divides
Isolating the last term instead of the first, we see that
ancn + an-1cn-1d + ... + a1cdn-1 = - a0dn
As before, since c is a factor of every term in the left side of this equation, c must divide a0dn. Since c and d are relatively prime, c and dn are relatively prime, and we conclude that c must divide a0.
Now consider the equation for the nth root of an integer t:
If r = c/d is a rational nth root of t expressed in lowest terms, the Rational Root Theorem states that d divides 1, the coefficient of xn. That is, that d must equal 1, and r = c must be an integer, and t must be itself a perfect nth power.
Of course, the same arguments could be applied to just the simple equation xn = t. (Indeed, such an argument is the essence of Proofs 2 and 4 of the irrationality of the square root of 2.) Somehow, though, the role of VII.24 and VII.30 is better seen in the more general case. And, for essentially the same effort, we get the useful Rational Root Theorem in the bargain.
(Note that Gauss' Lemma is a specialization of the above theorem to monic polynomials.)
Copyright © 1996-2018 Alexander Bogomolny