Binomials

We continue our study of binomials by determining conditions that characterize irreducibility and describing the Galois group of a

x

n

−

u

binomial in terms of 2 × 2 matrices over ℤ

n

. We then consider an application of binomials to determining the irrationality of linear combinations of radicals. Specifically, we prove that if

p

1

,....,

p

m

are distinct prime numbers, then the degree of

$$ \mathbb{Q}\left( {\sqrt[n]{{p_1 }},...,\sqrt[n]{{p_m }}} \right) $$

over ℚ is as large as possible, namely,

n

m

. This implies that the set of all products of the form

$$ \sqrt[n]{{p_1^{e\left( 1 \right)} }}...\sqrt[n]{{p_m^{e\left( m \right)} }} $$

where 0 ≤

e

(

i

) ≤

n

− 1, is linearly independent over ℚ For instance, the numbers

$$ 1,\sqrt[4]{3} = \sqrt[{60}]{{3^{15} }},\sqrt[5]{4} = \sqrt[{60}]{{2^{24} }}{\mathbf{ }}{\text{and}}{\mathbf{ }}\sqrt[6]{{72}} = \sqrt[{60}]{{2^{30} 3^{20} }} $$

are of this form, where

p

1

= 2,

p

2

= 3. Hence, any expression of the form

$$ a_1 \sqrt[4]{3} + a_2 \sqrt[5]{4} + a_3 \sqrt[6]{{72}} $$

where

a

i

∈ ℚ, must be irrational, unless

a

i

= 0 for all

i

.