The constant term of the minimal polynomial of over ℚ
© Demirci and Cangül; licensee Springer. 2013
Received: 21 January 2013
Accepted: 11 March 2013
Published: 29 March 2013
Let be the Hecke group associated to for integer. In this paper, we determine the constant term of the minimal polynomial of denoted by .
KeywordsHecke groups minimal polynomial constant term
where λ is a fixed positive real number.
These groups are isomorphic to the free product of two finite cyclic groups of orders 2 and q.
The first few Hecke groups are (the modular group), , , and . It is clear from the above that , but unlike in the modular group case (the case ), the inclusion is strict and the index is infinite as is discrete, whereas is not for .
Here we use Chebycheff polynomials.
For , Cangul denoted the minimal polynomial of over Q by . Then he obtained the following formula for the minimal polynomial .
Theorem 1 ([, Theorem 1])
If , then , and if , then .
- (b)If m is an odd prime, then(5)
- (c)If , then(6)
- (d)If m is even and is odd, then(7)
- (e)Let m be odd and let p be a prime dividing m. If , then(8)
For the first four Hecke groups Γ, , , and , we can find the minimal polynomial, denoted by , of over Q as , , , and , respectively. However, for , the algebraic number is a root of a minimal polynomial of degree ≥3. Therefore, it is not possible to determine for as nicely as in the first four cases. Because of this, it is easy to find and study with the minimal polynomial of instead of itself. The minimal polynomial of has been used for many aspects in the literature (see [5–8] and ).
between and .
In , when the principal congruence subgroups of for prime were studied, we needed to know whether the minimal polynomial of is congruent to 0 modulo p for prime p and also the constant term of it modulo p.
In this paper, we determine the constant term of the minimal polynomial of . We deal with odd and even q cases separately. Of course, this problem is easier to solve when q is odd.
2 The constant term of
Therefore we need to calculate the product on the right-hand side of (11). To do this, we need the following result given in .
Lemma 1 .
To compute c, we let in (17). If d is odd, then as by (20). So, we are only concerned with even d. Indeed, if q is odd, then the left-hand side at is equal to ±1. Therefore we have the following result.
Proof It follows from (19) and (20). □
Let us now investigate the case of even q. As , h must be odd. So, by a similar discussion, we get the following.
we can calculate c.
In fact the calculations show that there are three possibilities:
where are distinct odd primes and , .
Here we consider the first two cases and .
If we continue, we can find other divisors d of q, similarly. Finally, there is divisor of the form . Thus, the product of all coefficients d in the factors in the numerator is equal to the product of all coefficients e in the factors in the denominator implying . Therefore the proof is completed. □
Now we give an example for all possible even q cases.
Dedicated to Professor Hari M Srivastava.
Both authors are supported by the Scientific Research Fund of Uludag University under the project number F2012/15 and the second author is supported under F2012/19.
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