primitive, Gauss Lemma

**xianghu21** · January 18th 2009, 12:19 PM

Let $A$ be a commutative ring and let $A[x]$ be the ring of polynomials in an indeterminate $x$ , with coeﬃcients in $A$ . Let $f = a_0 + a_1x + \cdots + a_nx^n \in A[x]$ . $f$ is said to be primitive if $(a_0, a_1, \cdots, a_n)=(1)$ . Prove that if $f$ , $g \in A[x]$ , then $fg$ is primitive iff $f$ and $g$ are primitive.
[This is Atiyah-Macdonald #1.2d]
I am particularly confused with $\Leftarrow$ which I assume uses Gauss Lemma.

For the $\Rightarrow$ , I have:
Suppose that $fg$ is primitive. Now suppose that $f$ were not primitive.
Then $(a_0, a_1, \ldots, a_n) = (1)$ so there is a common factor to all of the terms. But then $fg$ would have a common factor as well. Thus $f$ is primitive, and so must $g$ be.

**ThePerfectHacker** · January 18th 2009, 01:22 PM

Originally Posted by xianghu21

Let $A$ be a commutative ring and let $A[x]$ be the ring of polynomials in an indeterminate $x$ , with coeﬃcients in $A$ . Let $f = a_0 + a_1x + \cdots + a_nx^n \in A[x]$ . $f$ is said to be primitive if $(a_0, a_1, \cdots, a_n)=(1)$ . Prove that if $f$ , $g \in A[x]$ , then $fg$ is primitive iff $f$ and $g$ are primitive.
[This is Atiyah-Macdonald #1.2d]
I am particularly confused with $\Leftarrow$ which I assume uses Gauss Lemma.

For the $\Rightarrow$ , I have:
Suppose that $fg$ is primitive. Now suppose that $f$ were not primitive.
Then $(a_0, a_1, \ldots, a_n) = (1)$ so there is a common factor to all of the terms. But then $fg$ would have a common factor as well. Thus $f$ is primitive, and so must $g$ be.

Let $f(x) = a_nx^n + ... + a_1x + a_0$ and let $g(x) = b_mx^m+...+b_1x+b_0$ . Define $h(x) = f(x)g(x)$ . Let $\pi$ be an irreducible in $R$ . Now if $a_p$ is smallest coefficient not divisible by $\pi$ and $b_q$ is smallest coefficient not dividible by $\pi$ then $c_{p+q}$ is smallest coefficient (in $h(x) = c_Nx^N + ... + c_0$ ) not divisible by $\pi$ .

Prove the result above. After that, Gauss lemma's follows.

**NonCommAlg** · January 18th 2009, 01:25 PM

Originally Posted by xianghu21

Let $A$ be a commutative ring and let $A[x]$ be the ring of polynomials in an indeterminate $x$ , with coeﬃcients in $A$ . Let $f = a_0 + a_1x + \cdots + a_nx^n \in A[x]$ . $f$ is said to be primitive if $(a_0, a_1, \cdots, a_n)=(1)$ . Prove that if $f$ , $g \in A[x]$ , then $fg$ is primitive iff $f$ and $g$ are primitive.
[This is Atiyah-Macdonald #1.2d]
I am particularly confused with $\Leftarrow$ which I assume uses Gauss Lemma.

For the $\Rightarrow$ , I have:
Suppose that $fg$ is primitive. Now suppose that $f$ were not primitive.
Then $(a_0, a_1, \ldots, a_n) = (1)$ so there is a common factor to all of the terms. But then $fg$ would have a common factor as well. Thus $f$ is primitive, and so must $g$ be.

let $f(x)=\sum_{i=0}^na_ix^i,$ and $g(x)=\sum_{j=0}^mb_jx^j.$ let $f(x)g(x)=\sum_{k=0}^{m+n}c_kx^k.$ first see that by the definition of the coefficients $c_k$ if $t_k \in R, \; 0 \leq k \leq m+n,$ then $\sum_{k=0}^{m+n}t_kc_k=\sum_{i=0}^nr_ia_i=\sum_{j= 0}^ms_jb_j,$ for

some $r_i, s_j \in R.$ so if $fg$ is primitive, clearly both $f$ and $g$ must be primitive too.

conversely, suppose $f$ and $g$ are primitive but $fg$ is not primitive. so $\sum_{k=0}^{m+n}Rc_k \subseteq \mathfrak{m}$ , for some maximal ideal $\mathfrak{m}$ of $R.$ now for any $z \in R$ let $z+\mathfrak{m}=\overline{z} \in \frac{R}{\mathfrak{m}}.$ so $\overline{c_k}=0, \; 0 \leq k \leq m+n.$ thus:

$\overline{f}(x)\overline{g}(x)=(\sum_{i=0}^{n}\ove rline{a_i}x^i)(\sum_{j=0}^{m}\overline{b_j}x^j)=\s um_{k=0}^{m+n}\overline{c_k}x^k=0.$ but $\frac{R}{\mathfrak{m}}[x]$ is an integral domain. thus either $\overline{f}(x)=0$ or $\overline{g}(x)=0,$ which means either $\sum_{i=0}^nRa_i \subseteq \mathfrak{m}$ or $\sum_{j=0}^mRb_j \subseteq \mathfrak{m}.$ contradiction!

**NonCommAlg** · January 18th 2009, 02:22 PM

Originally Posted by ThePerfectHacker

Let $f(x) = a_nx^n + ... + a_1x + a_0$ and let $g(x) = b_mx^m+...+b_1x+b_0$ . Define $h(x) = f(x)g(x)$ . Let $\pi$ be an irreducible in $R$ . Now if $a_p$ is smallest coefficient not divisible by $\pi$ and $b_q$ is smallest coefficient not dividible by $\pi$ then $c_{p+q}$ is smallest coefficient (in $h(x) = c_Nx^N + ... + c_0$ ) not divisible by $\pi$ .

Prove the result above. After that, Gauss lemma's follows.

be careful! the ring is not even an integral domain! the problem is not that straightforward.

**ThePerfectHacker** · January 18th 2009, 07:02 PM

Originally Posted by NonCommAlg

be careful! the ring is not even an integral domain! the problem is not that straightforward.

Yes, you are correct. I usually only deal with unique factorization domains.
I guess this proof can work for special types of domains.

**GaloisTheory1** · January 23rd 2009, 12:58 AM

Alternatively:

=> contrapositive. wlog suppose f is not primitive. let I be the proper ideal generated by the coefficients of f. It is easily seen that all the coefficients of fg lie in I so that fg is not primitive.

Other direction requires a rigorous argument like above.

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