Advanced book on Mathematics Olympiad

58 2 Algebra

185.Prove that for any distinct integersa 1 ,a 2 ,...,anthe polynomial

P(x)=(x−a 1 )(x−a 2 )···(x−an)− 1

cannot be written as a product of two nonconstant polynomials with integer coeffi-

cients.

186.Prove that for any distinct integersa 1 ,a 2 ,...,anthe polynomial

P(x)=(x−a 1 )^2 (x−a 2 )^2 ···(x−an)^2 + 1

cannot be written as a product of two nonconstant polynomials with integer coeffi-

cients.

187.Associate to a prime the polynomial whose coefficients are the decimal digits of the
prime (for example, for the prime 7043 the polynomial isP(z)= 7 x^3 + 4 x+3).
Prove that this polynomial is always irreducible overZ[x].

188.Letpbe a prime number of the form 4k+3,kan integer. Prove that for any positive
integern, the polynomial(x^2 + 1 )n+pis irreducible in the ringZ[x].

189.Letpbe a prime number. Prove that the polynomial

P(x)=xp−^1 + 2 xp−^2 + 3 xp−^3 +···+(p− 1 )x+p

is irreducible inZ[x].

190.LetP(x)be a monic polynomial inZ[x], irreducible over this ring, and such that
|P( 0 )|is not the square of an integer. Prove that the polynomialQ(x)defined by
Q(x)=P(x^2 )is also irreducible overZ[x].

2.2.6 Chebyshev Polynomials...................................

Thenth Chebyshev polynomialTn(x)expresses cosnθas a polynomial in cosθ. This
means thatTn(x)=cos(narccosx), forn≥0. These polynomials satisfy the recurrence

T 0 (x)= 1 ,T 1 (x)=x, Tn+ 1 (x)= 2 xTn(x)−Tn− 1 (x), forn≥ 1.

For example,T 2 (x)= 2 x^2 −1,T 3 (x)= 4 x^3 − 3 x,T 4 (x)= 8 x^4 − 8 x^2 +1.
One usually calls these the Chebyshev polynomials of the first kind, to distinguish
them from the Chebyshev polynomials of the second kindUn(x)defined byU 0 (x)= 1 ,
U 1 (x) = 2 x, Un+ 1 (x)= 2 xUn(x)−Un− 1 (x), forn ≥1 (same recurrence relation
but different initial condition). Alternatively,Un(x)can be defined by the equality
Un(cosθ)=sin(nsin+θ^1 )θ.