Abstact
We continue our study of binomials by determining conditions that characterize irreducibility and describing the Galois group of a xn − u binomial in terms of 2 × 2 matrices over ℤn. We then consider an application of binomials to determining the irrationality of linear combinations of radicals. Specifically, we prove that if p1,....,pm are distinct prime numbers, then the degree of
over ℚ is as large as possible, namely, nm. This implies that the set of all products of the form
where 0 ≤ e(i) ≤ n − 1, is linearly independent over ℚ For instance, the numbers
are of this form, where p1 = 2, p2 = 3. Hence, any expression of the form
where ai ∈ ℚ, must be irrational, unless ai = 0 for all i.
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© 2006 Springer New York
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(2006). Binomials. In: Field Theory. Graduate Texts in Mathematics, vol 158. Springer, New York, NY. https://doi.org/10.1007/0-387-27678-5_15
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DOI: https://doi.org/10.1007/0-387-27678-5_15
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-27677-9
Online ISBN: 978-0-387-27678-6
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