Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

APP. B] ALGEBRAIC SYSTEMS 449

Theorem B.21: Letfandgbe polynomials inK[t], not both zero. Then there exists a unique monic polynomial
dsuch that:
(i) ddivides bothfandg. (ii) Ifd′dividesfandg, then d′dividesd.
The polynomialdin the above Theorem B.21 is called thegreatest common divisoroffandg, written
d=gcd(f,g). Ifd=1, thenfandgare said to berelatively prime.

Corollary B.22: Letdbe the greatest common divisor offandg. Then there exist polynomialsmandnsuch
thatd=mf+ng. In particular, iffandgare relatively prime, then there exist polynomialsm
andnsuch thatmf+ng=1.
A polynomialp∈K[t]is said to beirreducibleifpis not a scalar and ifp=fgimpliesforgis a scalar.
In other words,pis irreducible if its only divisors are its associates (scalar multiples). The following lemma
(proved in Problem B.36) applies.
Lemma B.23:Supposep∈K[t]is irreducible. Ifpdivides the productfgof polynomialsfandginK[t],then
pdividesforpdividesg.More generally, ifpdivides the productf 1 f 2 ···fnofnpolynomials,
thenpdivides one of them.
The next theorem (proved in Problem B.37) states that the polynomials over a field form aunique factorization
domain(UFD).

Theorem B.24 (Unique Factorization Theorem): Letfbe a nonzero polynomial inK[t]. Thenfcan be written
uniquely (except for order) as a product

f=kp 1 p 2 ...pn

wherek∈Kand thep’s are monic irreducible polynomials inK[t].

Fundamental Theorem of Algebra

The proof of the following theorem lies beyond the scope of this text.

Fundamental Theorem of Algebra:Any nonzero polynomialf(t)over the complex fieldChas a root inC.

Thusf(t)can be written uniquely (except for order) as a product

f(t)=k(t−r 1 )(t−r 2 )···(t−rn)

wherekand theriare complex numbers and deg(f )=n.
The above theorem is certainly not true for the real fieldR. For example,f(t)=t^2 +1 is a polynomial
overR, butf(t)has no real root.
The following theorem (proved in Problem B.38) does apply.

Theorem B.25: Supposef(t)is a polynomial over the real fieldR, and suppose the complex number
z=a+bi, b=0, is a root off(t). Then the complex conjugatez ̄=a−biis also a root off(t).
Hence the following is a factor off(t):

c(t)=(t−z)(t− ̄z)=t^2 − 2 at+a^2 +b^2

The following theorem follows from Theorem B.25 and the Fundamental Theorem of Algebra.

Theorem B.26: Letf(t)be a nonzero polynomial over the real fieldR. Thenf(t)can be written uniquely
(except for order) as a product

f(t)=kp 1 (t)p 2 (t)···pn(t)

wherek∈Rand thepi(t) are real monic polynomials of degree 1 or 2.