Abstract Algebra - Polynomials: Irreducibles and Unique Factorization

VinnyCee

1. The problem statement, all variables and given/known data

Show that [tex]x^2\,+\,x[/tex] can be factored in two ways in [tex]\mathbb{Z}_6[x][/tex] as the product of nonconstant polynomials that are not units.


2. Relevant equations

Theorem 4.8

Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a unit in R.


Corollary 4.9

Let F be a field. then f(x) is a unit in F[x] if and only if f(x) is a nonzero constant polynomial.


Definition

Let F be a field. A nonconstant polynomial p(x) [itex]\in[/itex] F[x] is said to be irreducible if its only divisors are its associates and the nonzero constant polynomials (units). A nonconstant polynomial that is not irreducible is said to be reducible.


Theorem 4.10

Let f be a field. A nonzero polynomial f(x) is reducible in F[x] if and only if f(x) can be written as the product of two polynomials of lower degree.


Theorem 4.11

Let F be a field and p(x) a nonconstant polynomial in F[x]. then the following conditions are equivalent:

(1) p(x) is irreducible.

(2) If b(x) and c(x) are any polynomials such that p(x)|b(x) c(x), then p(x)|b(x) or p(x)|c(x).

(3) If r(x) and s(x) are any polynomials such that p(x) = r(x) s(x), then r(x) or s(x) is a nonzero constant polynomial.


Corollary 4.12

Let F be a field and p(x) an irreducible polynomial in F[x]. If [itex]p(x)|a_1(x)\,a_2(x)\,\cdots\,a_n(x)[/itex], then p(x) divides at least one of the [itex]a_i(x)[/itex].


NOTE: An element a in a commutative ring with identity R is said to be an associate of an element b of R if a = bu for some unit u. In the integer ring, the only associate of an integer n are n and -n because [itex]\pm\,1[/itex] are the only units.


3. The attempt at a solution

Obviously, x(x+1) is one of the ways to factor it, but what is the other?

Dick

Try and find another root of f(x)=x^2+x besides 0 and -1. That will tell you how to factor it.

VinnyCee

Well, since it is in [tex]\mathbb{Z}_6[/tex], I know that 6 is congruent to 0 mod 6. But what is congruent to -1 mod 6? -1 is not even in [tex]\mathbb{Z}_6[/tex], right?

[tex]6\,\equiv\,0\,\left(mod\,6\right)[/tex]

but

[tex]???\,\equiv\,-1\,\left(mod\,6\right)[/tex]

micromass

5 is congruent to -1 in (mod 6). But you'll need to find another root except 5 and 0...

VinnyCee

Not relevant to the problem above, but... here is the long division for problem 4.4.2(a)!



[tex]
\begin{equation*}
\begin{array}{rc@{}c}
& \multicolumn{2}{l}{\, \, \, x^9\,+\,x^8\,+\,2x^7\,+\,2x^6\,+\,2x^5\,+\,2x^4\,+\,2x^3\,+\,2x^2\,+\,2 x\,+\,2} \vspace*{0.12cm} \\ \cline{2-3}
\multicolumn{1}{r}{x\,-\,1 \hspace*{-4.8pt}} & \multicolumn{1}{l}{ \hspace*{-5.6pt} \Big) \hspace*{20pt} x^{10}\,+\,x^8} \\
& \multicolumn{2}{l}{ -\left(x^{10}\,-\,x^9\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{72pt}x^9\,+\,x^8}
& \multicolumn{2}{l}{\hspace*{98pt}-\left(x^9\,-\,x^8\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{108pt}2\,x^8}
& \multicolumn{2}{l}{\hspace*{134pt}-\left(2\,x^8\,-\,2\,x^7\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{164pt}2\,x^7}
& \multicolumn{2}{l}{\hspace*{190pt}-\left(2\,x^7\,-\,2\,x^6\right)} \\ \cline{2-3}

& \multicolumn{2}{l}{\hspace*{220pt}2\,x^6}
& \multicolumn{2}{l}{\hspace*{246pt}-\left(2\,x^6\,-\,2\,x^5\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{276pt}2\,x^5}
& \multicolumn{2}{l}{\hspace*{302pt}-\left(2\,x^5\,-\,2\,x^4\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{332pt}2\,x^4}
& \multicolumn{2}{l}{\hspace*{358pt}-\left(2\,x^4\,-\,2\,x^3\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{388pt}2\,x^3}
& \multicolumn{2}{l}{\hspace*{414pt}-\left(2\,x^3\,-\,2\,x^2\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{444pt}2\,x^2}
& \multicolumn{2}{l}{\hspace*{470pt}-\left(2\,x^2\,-\,2\,x\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{500pt}2\,x}
& \multicolumn{2}{l}{\hspace*{526pt}-\left(2\,x\,-\,2\right)} \\ \cline{2-3}
& \multicolumn{2}{l}{\hspace*{556pt}2}
\end{array}
\end{equation*} [/tex]


It cuts off at the end, but I think you can do the last two lines:))

VinnyCee

So, [tex]5\,\equiv\,-1\,\left(mod\,6\right)[/tex]...

How do I find another root?

micromass

There are only 6 elements in [tex]\mathbb{Z}_6[/tex]. I guess it's not too hard to check them all and see whether they are roots?

Dick

Yes, please. Check them all. You already know 0 and 5 work. So you don't need to check them. Now you have only four possibilities to check!

VinnyCee

1 and 2 produce [tex]x^2\,-\,3\,x\,+\,2[/tex] so that doesn't work.

3 and 4 produce [tex]x^2\,-\,7\,x\,+12\,\longrightarrow\,x^2\,-\,x[/tex] so that doesn't work.

1 and 3 produce [tex]x^2\,-\,4\,x\,+3[/tex] so that doesn't work.

1 and 4 produce [tex]x^2\,-\,5\,x\,+4[/tex] so that doesn't work.

2 and 3 produce [tex]x^2\,-\,5\,x\,+6\,\longrightarrow\,x^2\,-\,5\,x[/tex] so that doesn't work.

2 and 4 produce [tex]x^2\,-\,6\,x\,+8\,\longrightarrow\,x^2\,+\,2[/tex] so that doesn't work.

There are no other options that I can see!


Ohhhh.... wait a minute! What about 3 and -4?

[tex]\left(x\,-\,3\right)\,\left(x\,+\,4\right)\,=\,x^2\,+\,4\,x\,-\,3\,x\,-\,12\,=\,x^2\,+\,x[/tex]

So 0 and 5 are roots and 3 and -4 are roots. Is that the correct answer to this problem or is there something more general I should be writing in order to "show" this is true?

Dick

Sure. That's all you need. Nothing more general required. That's a different factorization than x*(x+1), yes? BTW you don't have to check them in pairs. If 2 is a root of x^2+x (which it is) then you know (x-2) is a factor. Then you can just divide to find the other factor.

micromass

By the way, 2 and 3 does give you the right answer since [tex]x^2-5x=x^2+x[/tex]. This follows since -5=1 (mod 6).