The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set
of different primes are linearly independent over the field of
rationals.

I’ve tried to solve the problem using elementary algebra
and also using the theory of field extensions, without success. To
prove linear independence of two primes is easy but then my problems
arise. I would be very thankful for an answer to this question.

Answer

Below is a simple proof from one of my old sci.math posts, followed by reviews of a few related papers.

Theorem $\ $ Let $\rm\,Q\,$ be a field with $2 \ne 0,\,$ and $\rm\ L = Q(S)\ $ be an extension of $\rm\,Q\,$ generated by $\rm\, n\,$ square roots $\rm\ S = \{ \sqrt{a}, \sqrt{b},\ldots \}$ of $\rm\ a,b,\,\ldots \in Q.\,$
If every nonempty subset of $\rm\,S\,$ has product $\rm\not\in Q\,$ then each successive
adjunction $\rm\ Q(\sqrt{a}),\ Q(\sqrt{a},\sqrt{b}),\,\ldots$ doubles degree over $\rm Q,\,$ so, in total, $\rm\, [L:Q] = 2^n.\,$ Thus the $\rm 2^n$ subproducts of the product of $\rm\,S\, $ are a basis of $\rm\,L\,$ over $\rm\,Q.$

Proof $\ $ By induction on the tower height $\rm\,n =$ number of root adjunctions. The Lemma below implies $\rm\ [1, \sqrt{a}\,]\ [1, \sqrt{b}\,] = [1, \sqrt{a}, \sqrt{b}, \sqrt{ab}\,]\ $ is a $\rm\,Q$-vector space basis of $\rm\, Q(\sqrt{a}, \sqrt{b})\ $ iff $\ 1\ $ is the only basis element in $\rm\,Q.\,$
We lift this to $\rm\, n > 2\,$ i.e. to $\, [1, \sqrt{a_1}\,]\ [1, \sqrt{a_2}\,]\cdots [1, \sqrt{a_n}\,]\,$ with $2^n$ elts.

$\rm n = 1\!:\ L = Q(\sqrt{a})\ $ so $\rm\,[L:Q] = 2,\,$ since $\rm\,\sqrt{a}\not\in Q\,$ by hypothesis.

$\rm n > 1\!:\ L = K(\sqrt{a},\sqrt{b}),\,\ K\ $ of height $\rm\,n\!-\!2.\,$ By induction $\rm\,[K:Q] = 2^{n-2} $ so we need only show $\rm\, [L:K] = 4,\,$ since then $\rm\,[L:Q] = [L:K]\ [K:Q] = 4\cdot 2^{n-2}\! = 2^n.\,$ The lemma below shows $\rm\,[L:K] = 4\,$ if $\rm\ r = \sqrt{a},\ \sqrt{b},\ \sqrt{a\,b}\ $ all $\rm\not\in K,\,$
true by induction on $\rm\,K(r)\,$ of height $\rm\,n\!-\!1\,$ shows $\rm\,[K(r):K] = 2\,$ $\Rightarrow$ $\rm\,r\not\in K.\quad$ QED

Lemma $\rm\ \ [K(\sqrt{a},\sqrt{b}) : K] = 4\ $ if $\rm\ \sqrt{a},\ \sqrt{b},\ \sqrt{a\,b}\ $ all $\rm\not\in K\,$ and $\rm\, 2 \ne 0\,$ in $\rm\,K.$

Proof $\ \ $ Let $\rm\ L = K(\sqrt{b}).\,$ $\rm\, [L:K] = 2\,$ by $\rm\,\sqrt{b} \not\in K,\,$ so it suffices to show $\rm\, [L(\sqrt{a}):L] = 2.\,$ This fails only if $\rm\,\sqrt{a} \in L = K(\sqrt{b})$ $\,\Rightarrow\,$ $\rm \sqrt{a}\ =\ r + s\ \sqrt{b}\ $ for $\rm\ r,s\in K,\,$ which is false, because squaring yields $\rm\,(1):\ \ a\ =\ r^2 + b\ s^2 + 2\,r\,s\ \sqrt{b},\, $ which is contra to hypotheses as follows:

$\rm\qquad\qquad rs \ne 0\ \ \Rightarrow\ \ \sqrt{b}\ \in\ K\ \ $ by solving $(1)$ for $\rm\sqrt{b},\,$ using $\rm\,2 \ne 0$

$\rm\qquad\qquad\ s = 0\ \ \Rightarrow\ \ \ \sqrt{a}\ \in\ K\ \ $ via $\rm\ \sqrt{a}\ =\ r + s\ \sqrt{b}\ =\ r \in K$

$\rm\qquad\qquad\ r = 0\ \ \Rightarrow\ \ \sqrt{a\,b}\in K\ \ $ via $\rm\ \sqrt{a}\ =\ s\ \sqrt{b},\, \ $times $\rm\,\sqrt{b}\quad\quad$ QED

In the classical case $\rm\:Q\:$ is the field of rationals and the square roots
have radicands being distinct primes. Here it is quite familiar that a
product of any nonempty subset of them is irrational since, over a UFD,
a product of coprime elements is a square iff each factor is a square
(mod units). Hence the classical case satisfies the theorem’s hypotheses.

Elementary proofs like that above are often credited to Besicovitch
(see below). But I have not seen his paper so I cannot say for sure
whether or not Besicovic’s proof is essentially the same as above.
Finally, see the papers reviewed below for some stronger results.


2,33f 10.0X
Besicovitch, A. S.
On the linear independence of fractional powers of integers.
J. London Math. Soc. 15 (1940). 3-6.

Let $\ a_i = b_i\ p_i,\ i=1,\ldots s\:,\:$ where the $p_i$ are $s$ different primes and
the $b_i$ positive integers not divisible by any of them. The author proves
by an inductive argument that, if $x_j$ are positive real roots of
$x^{n_j} – a_j = 0,\ j=1,…,s ,$ and $P(x_1,…,x_s)$ is a polynomial with
rational coefficients and of degree not greater than $n_j – 1$ with respect
to $x_j,$ then $P(x_1,…,x_s)$ can vanish only if all its coefficients vanish. $\quad$ Reviewed by W. Feller.


15,404e 10.0X
Mordell, L. J.
On the linear independence of algebraic numbers.
Pacific J. Math. 3 (1953). 625-630.

Let $K$ be an algebraic number field and $x_1,\ldots,x_s$ roots of the equations
$\ x_i^{n_i} = a_i\ (i=1,2,…,s)$ and suppose that (1) $K$ and all $x_i$ are real, or
(2) $K$ includes all the $n_i$ th roots of unity, i.e. $ K(x_i)$ is a Kummer field.
The following theorem is proved. A polynomial $P(x_1,…,x_s)$ with coefficients
in $K$ and of degrees in $x_i$, less than $n_i$ for $i=1,2,\ldots s$, can vanish only if
all its coefficients vanish, provided that the algebraic number field $K$ is such
that there exists no relation of the form $\ x_1^{m_1}\ x_2^{m_2}\:\cdots\: x_s^{m_s} = a$, where $a$ is a number in $K$ unless $\ m_i \equiv 0 \mod n_i\ (i=1,2,…,s)$. When $K$ is of the second type, the theorem was proved earlier by Hasse [Klassenkorpertheorie,
Marburg, 1933, pp. 187–195] by help of Galois groups. When $K$ is of the first
type and $K$ also the rational number field and the $a_i$ integers, the theorem was proved by Besicovitch in an elementary way. The author here uses a proof analogous to that used by Besicovitch [J. London Math. Soc. 15b, 3–6 (1940) these Rev. 2, 33]. $\quad$ Reviewed by H. Bergstrom.


46 #1760 12A99
Siegel, Carl Ludwig
Algebraische Abhaengigkeit von Wurzeln. (German)
Acta Arith. 21 (1972), 59-64.

Two nonzero real numbers are said to be equivalent with respect to a real
field $R$ if their ratio belongs to $R$. Each real number $r \ne 0$ determines
a class $[r]$ under this equivalence relation, and these classes form a
multiplicative abelian group $G$ with identity element $[1]$. If $r_1,\dots,r_h$
are nonzero real numbers such that $r_i^{n_i}\in R$ for some positive integers $n_i\
(i=1,…,h)$
, denote by $G(r_1,…,r_h) = G_h$ the subgroup of $G$ generated by
$[r_1],\dots,[r_h]$ and by $R(r_1,…,r_h) = R_h$ the algebraic extension field of
$R = R_0$ obtained by the adjunction of $r_1,…,r_h$. The central problem
considered in this paper is to determine the degree and find a basis of $R_h$
over $R$. Special cases of this problem have been considered earlier by A. S.
Besicovitch [J. London Math. Soc. 15 (1940), 3-6; MR 2, 33] and by L. J.
Mordell [Pacific J. Math. 3 (1953), 625-630; MR 15, 404]. The principal
result of this paper is the following theorem: the degree of $R_h$ with respect
to $R_{h-1}$ is equal to the index $j$ of $G_{h-1}$ in $G_h$, and the powers $r_i^t\
(t=0,1,…,j-1)$
form a basis of $R_h$ over $R_{h-1}$. Several interesting
applications and examples of this result are discussed. $\quad$ Reviewed by H. S. Butts

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Source : Link , Question Author : user8465 , Answer Author : Bill Dubuque

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