Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

HERENDI, T. and PETHŐ, A., Trinomials, which are divisible by quadratic polynomials

68 Tamás Herendi and Attila Pethő Lemma 7. Let suppose that gcd(n, k) = m. Then gcd (j,""

1 2 ./„_! (X, J/

2) , /1 ™

d 2 • («, /)) = PROOF. By Lemma 2 (a) we know that F

n(x) has [f] différent real roots. Let suppose that they are Ii,

..., £r„j. Then [f] F

n(x) = le (F

n) JJ (x - Xi)

, i=

1 where le (F

n) is the leading coefficient of F

n

y which is 1 if n is even and n +1 if n is odd. Then by Lemma 6 W yn mod 2 . /„ y

2) = le (F„) • H (* - W) ' m0<

i i = 1 It is clear that (x — Xj • y

2) is irreducible, and by the unique factorization in a polynomial ring, this is the only possible factorization of y

n mod 2 • f

n {x,y

2). By Lemma 2(a) {x - t • y

2) \ y

n~

l mod

2 •f

n_

x (x,y

2) if and only if there exists j E {l,

..., [y] } such that t = — • Of course, then for ah Í, conjugate of t, (x - t • y

2) | y

n~

l mod

2 • f

n_

x {x,y

2). If t is such that (x-«V]ir

1,DDÍ

,-/n-i M and (x - t- y

2) \ y

k~

l mod

2 • /

f

c_ 1 (x, y

2) then there exist 6 {1,..., [f] } such that , .

2 = . ^ x 2 from where we get either Û. — P, or £1 = (£»+i) (Ci+i) 0 Sn s/c S

n . Without loss of generality we can suppose that £

J

n = It is easy to see, if m = gcd(n,k) then (tó)™ = (Ck)

m

, which means that ££ is mth root of unity and so (x - t • y

2) | y

m~

l mod 2 • f

m_

x (x,y

2). Eeversing, if (x - t • y

2) I y

m1 mod 2 • /

m_ 1 (x, y

2) then there exists i G {l,

..., [f] } such that t = — and if m | n then there exists j 6 {l,

..., [f] } such that Cm = tó o

r Cm = Çn'\ whic

h means that (x - t • y

2) | y

n~

l mod

2 • f

n-i (x

, y

2). We have m — 1 mod 2 = 1 if and only if n — 1 mod 2 = 1 and k - 1 mod 2 = 1. So if y | y

n~

l mod 2 • f

n_

x (x, y

2) and y | y*"

1 mod 2 •