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Section 6.4 Factorization in \(D\lbrack x\rbrack\)

One of the most important polynomial rings is \({\mathbb Z}[x]\text{.}\) One of the first questions that come to mind about \({\mathbb Z}[x]\) is whether or not it is a UFD. We will prove a more general statement here.
Let \(D\) be a unique factorization domain and suppose that
\begin{equation*} p(x) = a_n x^n + \cdots + a_1 x + a_0 \end{equation*}
in \(D[x]\text{.}\) Then the content of \(p(x)\) is the greatest common divisor of \(a_0, \ldots, a_n\text{.}\) We say that \(p(x)\) is primitive if \(\gcd(a_0, \ldots, a_n ) = 1\text{.}\)

Example 6.26.

In \({\mathbb Z}[x]\) the polynomial \(p(x)= 5 x^4 - 3 x^3 + x -4\) is a primitive polynomial since the greatest common divisor of the coefficients is 1; however, the polynomial \(q(x) = 4 x^2 - 6 x + 8\) is not primitive since the content of \(q(x)\) is 2.

Proof.
Let \(f(x) = \sum_{i=0}^{m} a_i x^i\) and \(g(x) = \sum_{i=0}^{n} b_i x^i\text{.}\) Suppose that \(p\) is a prime dividing the coefficients of \(f(x) g(x)\text{.}\) Let \(r\) be the smallest integer such that \(p \notdivide a_r\) and \(s\) be the smallest integer such that \(p \notdivide b_s\text{.}\) The coefficient of \(x^{r+s}\) in \(f(x) g(x)\) is
\begin{equation*} c_{r + s} = a_0 b_{r + s} + a_1 b_{r + s - 1} + \cdots + a_{r + s - 1} b_1 + a_{r + s} b_0. \end{equation*}
Since \(p\) divides \(a_0, \ldots, a_{r-1}\) and \(b_0, \ldots, b_{s-1}\text{,}\) \(p\) divides every term of \(c_{r+s}\) except for the term \(a_r b_s\text{.}\) However, since \(p \mid c_{r+s}\text{,}\) either \(p\) divides \(a_r\) or \(p\) divides \(b_s\text{.}\) But this is impossible.

Proof.
Let \(p(x) = c p_1(x)\) and \(q(x) = d q_1(x)\text{,}\) where \(c\) and \(d\) are the contents of \(p(x)\) and \(q(x)\text{,}\) respectively. Then \(p_1(x)\) and \(q_1(x)\) are primitive. We can now write \(p(x) q(x) = c d p_1(x) q_1(x)\text{.}\) Since \(p_1(x) q_1(x)\) is primitive, the content of \(p(x) q(x)\) must be \(cd\text{.}\)

Proof.
Let \(a\) and \(b\) be nonzero elements of \(D\) such that \(a f(x), b g(x)\) are in \(D[x]\text{.}\) We can find \(a_1, b_2 \in D\) such that \(a f(x) = a_1 f_1(x)\) and \(b g(x) = b_1 g_1(x)\text{,}\) where \(f_1(x)\) and \(g_1(x)\) are primitive polynomials in \(D[x]\text{.}\) Therefore, \(a b p(x) = (a_1 f_1(x))( b_1 g_1(x))\text{.}\) Since \(f_1(x)\) and \(g_1(x)\) are primitive polynomials, it must be the case that \(ab \mid a_1 b_1\) by Gauss’s Lemma. Thus there exists a \(c \in D\) such that \(p(x) = c f_1(x) g_1(x)\text{.}\) Clearly, \(\deg f(x) = \deg f_1(x)\) and \(\deg g(x) = \deg g_1(x)\text{.}\)
The following corollaries are direct consequences of Lemma 6.29.

Proof.
Let \(p(x)\) be a nonzero polynomial in \(D[x]\text{.}\) If \(p(x)\) is a constant polynomial, then it must have a unique factorization since \(D\) is a UFD. Now suppose that \(p(x)\) is a polynomial of positive degree in \(D[x]\text{.}\) Let \(F\) be the field of fractions of \(D\text{,}\) and let \(p(x) = f_1(x) f_2(x) \cdots f_n(x)\) by a factorization of \(p(x)\text{,}\) where each \(f_i(x)\) is irreducible. Choose \(a_i \in D\) such that \(a_i f_i(x)\) is in \(D[x]\text{.}\) There exist \(b_1, \ldots, b_n \in D\) such that \(a_i f_i(x) = b_i g_i(x)\text{,}\) where \(g_i(x)\) is a primitive polynomial in \(D[x]\text{.}\) By Corollary 6.30, each \(g_i(x)\) is irreducible in \(D[x]\text{.}\) Consequently, we can write
\begin{equation*} a_1 \cdots a_n p(x) = b_1 \cdots b_n g_1(x) \cdots g_n(x). \end{equation*}
Let \(a = a_1 \cdots a_n\text{.}\) Since \(g_1(x) \cdots g_n(x)\) is primitive, \(a\) divides \(b_1 \cdots b_n\text{.}\) Therefore, \(p(x) = a g_1(x) \cdots g_n(x)\text{,}\) where \(a \in D\text{.}\) Since \(D\) is a UFD, we can factor \(a\) as \(u c_1 \cdots c_k\text{,}\) where \(u\) is a unit and each of the \(c_i\)’s is irreducible in \(D\text{.}\)
We will now show the uniqueness of this factorization. Let
\begin{equation*} p(x) = a_1 \cdots a_m f_1(x) \cdots f_n(x) = b_1 \cdots b_r g_1(x) \cdots g_s(x) \end{equation*}
be two factorizations of \(p(x)\text{,}\) where all of the factors are irreducible in \(D[x]\text{.}\) By Corollary 6.30, each of the \(f_i\)’s and \(g_i\)’s is irreducible in \(F[x]\text{.}\) The \(a_i\)’s and the \(b_i\)’s are units in \(F\text{.}\) Since \(F[x]\) is a UFD, \(n=s\text{.}\) Now rearrange the \(g_i(x)\)’s so that \(f_i(x)\) and \(g_i(x)\) are associates for \(i = 1, \ldots, n\text{.}\) Then there exist \(c_1, \ldots, c_n\) and \(d_1, \ldots, d_n\) in \(D\) such that \((c_i / d_i) f_i(x) = g_i(x)\) or \(c_i f_i(x) = d_i g_i(x)\text{.}\) The polynomials \(f_i(x)\) and \(g_i(x)\) are primitive; hence, \(c_i\) and \(d_i\) are associates in \(D\text{.}\) Thus, \(a_1 \cdots a_m = u b_1 \cdots b_r\) in \(D\text{,}\) where \(u\) is a unit in \(D\text{.}\) Since \(D\) is a unique factorization domain, \(m = s\text{.}\) Finally, we can reorder the \(b_i\)’s so that \(a_i\) and \(b_i\) are associates for each \(i\text{.}\) This completes the uniqueness part of the proof.
The theorem that we have just proven has several obvious but important corollaries.

Subsection Historical Note

Karl Friedrich Gauss, born in Brunswick, Germany on April 30, 1777, is considered to be one of the greatest mathematicians who ever lived. Gauss was truly a child prodigy. At the age of three he was able to detect errors in the books of his father’s business. Gauss entered college at the age of 15. Before the age of 20, Gauss was able to construct a regular \(17\)-sided polygon with a ruler and compass. This was the first new construction of a regular \(n\)-sided polygon since the time of the ancient Greeks. Gauss succeeded in showing that if \(N= 2^{2^n} + 1\) was prime, then it was possible to construct a regular \(N\)-sided polygon.
Gauss obtained his Ph.D. in 1799 under the direction of Pfaff at the University of Helmstedt. In his dissertation he gave the first complete proof of the Fundamental Theorem of Algebra, which states that every polynomial with real coefficients can be factored into linear factors over the complex numbers. The acceptance of complex numbers was brought about by Gauss, who was the first person to use the notation of \(i\) for \(\sqrt{-1}\text{.}\)
Gauss then turned his attention toward number theory; in 1801, he published his famous book on number theory, Disquisitiones Arithmeticae. Throughout his life Gauss was intrigued with this branch of mathematics. He once wrote, “Mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”
In 1807, Gauss was appointed director of the Observatory at the University of Göttingen, a position he held until his death. This position required him to study applications of mathematics to the sciences. He succeeded in making contributions to fields such as astronomy, mechanics, optics, geodesy, and magnetism. Along with Wilhelm Weber, he coinvented the first practical electric telegraph some years before a better version was invented by Samuel F. B. Morse.
Gauss was clearly the most prominent mathematician in the world in the early nineteenth century. His status naturally made his discoveries subject to intense scrutiny. Gauss’s cold and distant personality many times led him to ignore the work of his contemporaries, making him many enemies. He did not enjoy teaching very much, and young mathematicians who sought him out for encouragement were often rebuffed. Nevertheless, he had many outstanding students, including Eisenstein, Riemann, Kummer, Dirichlet, and Dedekind. Gauss also offered a great deal of encouragement to Sophie Germain (1776–1831), who overcame the many obstacles facing women in her day to become a very prominent mathematician. Gauss died at the age of 78 in Göttingen on February 23, 1855.
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