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Might $a(x)$ be irreducible in $F[x]$ but reducible in $F[x]/\langle p(x)\rangle$, where $a(x)$ is not a multiple of $p(x)$?

$F[x]/\langle p(x)\rangle $ does not have to be a field. Could you name an example, preferably one for a field another for a non-field just so I can see the difference?

It was suggested by Arturo that I should re-edit this question since it was not clear enough: Let F be a field, and say $a(x)∈F[x]$ be irreducible. If $p(x)$ is not constant and not equal to an associate of $a(x)$, and we let $R=F[x]/(p(x))$, and identify F with its image in R, can $a(y)$ be reducible in $R[y]$?

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I think there may be a problem with the question you've posed. I assume you mean $\mathbb{F}$ is a field. Correct? If so, $\mathbb{F}[x]$ is a principal ideal domain and as such every non-trivial quotient has the property that every non-zero element is either a zero divisor or a unit. Thus if a quotient lacks zero divisors, it's a field. We usually only speak of irreducibles in the context of an integral domain. So in your context speaking of a reducible/irreducible element only makes sense if the quotient is also a field. In this case, there are no irreducibles (only units and zero). –  Bill Cook Oct 22 '11 at 2:08
    
Oops! I guess I read the title of your question and not the contents. My comment addresses the issue with your title. I'll give you an example addressing your question below. –  Bill Cook Oct 22 '11 at 2:14
    
I understand your first comment. It was helpful. –  crazy student Oct 22 '11 at 2:44
    
Please make sure the question you ask is actually in the post, not just in the title. –  Arturo Magidin Oct 22 '11 at 3:27
    
@Bill: Actually, it is perfectly sensible to talk about irreducible and prime elements in the context of any (commutative) ring; on the other hand, talking about reducible elements (and reducible polynomials) is a rather different kettle of fish. –  Arturo Magidin Oct 22 '11 at 4:10

3 Answers 3

up vote 1 down vote accepted

First, let's be clear:

If $R$ is a commutative ring with $1$, then an element $a\in R$ is irreducible if and only if $a$ is not a unit, $a$ is not a zero divisor, and whenever $a=bc$ in $R$, either $b$ is a unit (invertible) or $c$ is a unit. Added. Elements are not usually called "reducible", though. Elements that are not irreducible are "not irreducible".

In the particular case of $R=F[x]$ with $F$ a field, this agrees with the "usual" notion of "irreducible polynomial", since the units are precisely the nonzero constant polynomials.

Now, if $F$ is a field, then $F[x]$ is a unique factorization domain (even more, a Euclidean domain). If $a(x)$ is irreducible, then it is a prime in $F[x]$. So for any $p(x)\in F[x]$ there are only three possibilities: either $a(x)$ divides $p(x)$, or $p(x)=0$, or else $a(x)$ and $p(x)$ are relatively prime, and in particular, there exist polynomials $r(x)$ and $s(x)$ such that $$a(x)r(x) + p(x)s(x) = 1.$$

Now, suppose that $p(x)$ is a polynomial. The only way for $a(x)$ to be a multiple of $p(x)$ is for $a(x)$ to be of the form $a(x)=\lambda p(x)$ for some nonzero constant $\lambda\in F$. Assume that $a(x)$ is not a multiple of $p(x)$.

If $p(x)\neq 0$ and $a(x)$ divides $p(x)$, then $p(x) = a(x)q(x)$ for some $q(x)\notin F$. We cannot have $p(x)$ dividing $q(x)$, so then $\overline{q(x)}$ is not zero in $F[x]/\langle p(x)\rangle$. Neither is $\overline{a(x)}$. Yet $$\overline{a(x)}\times\overline{q(x)} = \overline{a(x)q(x)} = \overline{p(x)} = 0.$$ So $\overline{a(x)}$ is not irreducible, because it is a zero divisor.

If $a(x)$ does not divide $p(x)$, then we can write $1 = a(x)r(x)+p(x)s(x)$ for some polynomials $r(x)$ and $s(x)$. But then in $F[x]/\langle p(x)\rangle$ we have: $$1 = \overline{1} = \overline{a(x)r(x) + p(x)s(x)} = \overline{a(x)}\overline{r(x)} + \overline{p(x)}\overline{s(x)} = \overline{a(x)}\overline{r(x)},$$ so $\overline{a(x)}$ is not irreducible because it is a unit.

Finally, if $p(x)=0$, then $F[x]/\langle p(x)\rangle$ is essentially just $F[x]$ itself again, so $a(x)$ is irreducible in the image because it's just $a(x)$ in $F[x]$.

So, to answer the question that was posed, if $p(x)\neq 0$ then $a(x)$ will not be irreducible in $F[x]/\langle p(x)\rangle$ no matter what.


That said...

But your use of the word "reducible" suggests that you are actually wondering about something else: whether "as a polynomial" it can be written as a product of two nonconstant polynomials, or perhaps whether "it has a root" when considered in $F[x]/\langle p(x)\rangle$.

In other words: say $p(x)$ is a nonconstant polynomial in $F[x]$. Then $R=F[x]/\langle p(x)\rangle$ is a ring that contains a copy of $F$. We can then consider the polynomial $a(x)$ as having coefficients in $R$ and ask whether it is reducible or irreducible over $R$. However, using $x$ here is confusing because $x$ is already playing a role in the definition of $R$, so it's better to switch to a different letter in order to talk about "polynomials with coefficients in $R$". So let's use $y$ to denote polynomials with coefficients in $R$, and so we will be asking whether $a(y)$ is irreducible over $R$ or reducible over $R$.

If $p(x)$ is irreducible over $F$, then $F[x]/\langle p(x)\rangle$ is a field that contains $F$, and you can consider $a(y)$ as a polynomial in $(F[x]/\langle p(x)\rangle[y]=R[y]$. There, it may be that $a(y)$ is irreducible or that it is reducible.

For an example where it is reducible, take $a(x) = x^4 + 1$ in $\mathbb{Q}[x]$, and $p(x) = x^2+1$, both irreducible. Since $\mathbb{Q}[x]/\langle p(x)\rangle = \mathbb{Q}(i)$, the rational complex numbers, in $\mathbb{Q}(i)[y]$ we have $a(y) = y^4+1 = (y^2+i)(y^2-i)$, reducible.

For an example where it is irreducible, take $a(x) = x^2 - 2$ and $p(x) = x^2-3$ in $\mathbb{Q}[x]$. Then $\mathbb{Q}[x]/\langle p(x)\rangle$ is just $\mathbb{Q}(\sqrt{3})$, and $a(y)$ has no roots in $\mathbb{Q}(\sqrt{3})$; being degree $2$, it is irreducible in $\mathbb{Q}(\sqrt{3})[y]$.

If $p(x)$ is reducible, then you end up with a ring that is not a field and has zero divisors. In any case, as a consequence of the Chinese Remainder Theorem, if we factor $p(x)$ into irreducibles, $$p(x) = q_1(x)^{\alpha_1}\cdots q_m(x)^{\alpha_m},$$ with $q_i$ and $q_j$ coprime for $i\neq j$, $\alpha_i\gt 0$, then $$R=F[x]/\langle p(x)\rangle \cong \frac{F[x]}{\langle q_1(x)^{\alpha_1}\rangle}\times\cdots\times \frac{F[x]}{\langle q_m(x)^{\alpha_m}\rangle},$$ and this $a(y)$ may or may not be irreducible over this ring.

For an example where it is reducible, take the example above in which $a(y)$ was reducible, but replace $p(x)$ with $(x^4+1)(x^2+2)$ or some other reducible polynomial that has $x^4+1$ as a factor. Then $a(y)$ will have a root in the quotient, because it will have a root in one of the factors. Actually, since $F$ is embedded into the quotient "diagonally", you need two irreducible polynomials over which $a(x)$ was reducible, or a power of one over which $a(x)$ was reducible. For an example where it is irreducible, do the same but now take the product of two polynomials over which $a(y)$ was irreducible. you can take $F=\mathbb{Q}$, $a(x) = x^2-2$, $p(x) = (x^2-3)(x^2-5)$. Then $F[x]/\langle p(x)\rangle \cong \mathbb{Q}(\sqrt{3})\times\mathbb{Q}(\sqrt{5})$, and that ring does not have a square root of $2$ either, so $a(y)$ is still irreducible.

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I have a question about your a(x) = x^4 + 1 and p(x) = x^2 + 1. Aren't we modding out by p(x) so x^2 is really congruent to -1 (mod p(x))? So x^4 + 1 would be (x^2)(x^2) + 1 = 1 + 1 = 2? What's the y supposed to be, I'm kind of confused about that notation. –  crazy student Oct 22 '11 at 4:37
    
@crazystudent: The problem is that your entire question is confused. Are you talking about "irreducibility of polynomials", or "irreducibility of elements"? If you are talking about elements, I answered the question in its entirety above the line. Since $x^4+1$ and $x^2+1$ are relatively prime, then $x^4+1$ becomes a unit in the quotient, so it cannot be irreducible (or "reducible", either). (cont) –  Arturo Magidin Oct 22 '11 at 4:41
    
@crazystudent: If you are talking about polynomials, then you should not use $a(x)$ for the polynomial in the quotient, because $x$ represents something else. You would really be asking whether the polynomial $y^4+1$ is irreducible over $\mathbb{Q}[x]/(p(x))=\mathbb{Q}[i]$ or whether it is reducible. But if you really want to talk about irreducibe/reducible elements, then (i) You need to be clear; (ii) you need to distinguish the element of $F[x]$ with its image in the quotient; (iii) you need to clarify your notation, which is confused right now. –  Arturo Magidin Oct 22 '11 at 4:43
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@crazystudent: Then what you wrote is your question is nonsense. First, the correct preposition would be "irreducible over", not "irreducible in". Second, $a(x)$ is not a polynomial over $F[x]/\langle p(x)\rangle$. That's why I switched letters. If you let $R=F[x]/\langle p(x)\rangle$, then you can consider polynomials with coefficients in $R$. But using $x$ for the variable is very bad form and potentially confusing, because $x$ already is playing a different role. So instead I called the variable $y$, and talk about polynomials in $R[y]$ being irreducible. –  Arturo Magidin Oct 22 '11 at 4:50
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@crazystudent: Here's the phrasing you seem to be going for: I suggest editing your question and adding this below your current phrasing. "Let $F$ be a field, and say $a(x)\in F[x]$ be irreducible. If $p(x)$ is not constant and not equal to $a(x)$, and we let $R=F[x]/(p(x))$, and identify $F$ with its image in $R$, can $a(y)$ be reducible in $R[y]$?" Note that we don't look at the image of $a(x)$ in $R$, we are looking at the same polynomial, but now think of it as having coefficients in $R$. –  Arturo Magidin Oct 22 '11 at 4:53

If you quotient by a reducible polynomial, you'll end up with zero divisors.

Consider $\mathbb{Q}[x]/(x^2-1)$. I'll refer to cosets "$f(x)+(x^2+1)$" just by their representatives "$f(x)$". Notice that $x+1,x-1$ are non-zero in this quotient, but $(x+1)(x-1)=x^2-1=0$ (mod $x^2-1$). So we've got zero divisors.

If you want a better feel for what this ring "really" looks like. It's not hard to show that $\mathbb{Q}[x]/(x^2-1) \cong \mathbb{Q}[x]/(x-1) \times \mathbb{Q}[x]/(x+1) \cong \mathbb{Q} \times \mathbb{Q}$. [Use the map $f(x) \mapsto (f(x)+(x-1),f(x)+(x+1))$ it's not hard to show this map is a homomorphism from $\mathbb{Q}[x]$ to $\mathbb{Q}[x]/(x-1) \times \mathbb{Q}[x]/(x+1)$ has kernel $(x^2-1)$ and using Chinese remaindering you can show it's onto. Then the result follows form the 1st isomorphism theorem.]

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Small typo on the second line :) –  Alexei Averchenko Oct 22 '11 at 2:31
    
Chinese remaindering? I don't follow you when you say that Bill. I'm only taking an introductory abstract algebra course –  crazy student Oct 22 '11 at 2:37
    
I just want a real example like functions a(x), p(x) and any ring F[x] –  crazy student Oct 22 '11 at 2:40

I will address what I believe you are trying to ask for in the first scenario.

Let $F = \mathbb{Q}$, $a(x) = x^4-5$, $p(x) = x^2-5$. Note that both $a(x)$ and $p(x)$ are both irreducible by Eisenstein criterion. But $F[x]/(p(x)) = F[\sqrt{5}] = F(\sqrt{5})$, and over the polynomial ring $F(\sqrt{5})[x]$ we can factor $a(x) = (x^2-\sqrt{5})(x^2+\sqrt{5})$.

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Maybe I didn't get the memo here but it seems like everyone is saying x^4 - 5= (x^2 - sqrt(5))(x^2 + sqrt(5)) without mentioning that x^2 = 5 (mod p(x)). That is the only part that doesn't make sense to me. It's like everyone is ignoring that fact. –  crazy student Oct 22 '11 at 4:45
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@crazystudent: You need to make up your mind. What you are doing is viewing $a(x)$ as an element of the quotient, not as a polynomial. If you want to talk about irreducibility of polynomials, as you claim, then you do not view $a$ as an element of the quotient, you view it as a polynomial with coefficients in the quotient. That's why what you are writing is nonsensical: the $x$ in $a$ is an indeterminate, not an element of the quotient. Nobody is "ignoring that fact", we are trying to make sense of your confusion. –  Arturo Magidin Oct 22 '11 at 5:03

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