McDaniel's restriction on odd perfect numbers and use of cyclotomic polynomials

Question

In 1969, McDaniel showed that an odd number of the form

$$N=p^\alpha\prod_{j=1}^tp_j^{2\beta_j},$$

where $p$, $p_1$, $p_2$, $\ldots$, $p_t$ are distinct primes and

$$p\equiv\alpha\equiv1\pmod4,$$

cannot be perfect if

$$2\beta_1\equiv2\beta_2\equiv_\,\ldots\,\equiv2\beta_t\equiv1\pmod3.$$

Unfortunately, he used cyclotomic polynomials - which I don't understand yet - to prove his claim. That's why I wonder if this can be proved without involving cyclotomic polynomials, or if their meaning can be explained in this context.

Answer 1

This article might be useful to you:

http://galloty.chez.com/papers/cyclotomic.pdf

Answer 2

The following papers may prove equally useful:

Necessary Conditions For the Non-existence of Odd Perfect Numbers - Jonathan Pearlman, May 23, 2005

NONEXISTENCE OF ODD PERFECT NUMBERS OF A CERTAIN FORM - Ronald Evans and Jonathan Pearlman, July 2005