dooporpplauttssg
2022-05-02
Answered

Prove that $\mathbb{Z}+\left(3x\right)$ is a subring of $\mathbb{Z}\left[x\right]$ and there is no surjective homomorphism from $\mathbb{Z}\left[x\right]\Rightarrow \mathbb{Z}+\left(3x\right)$

You can still ask an expert for help

Norah Small

Answered 2022-05-03
Author has **12** answers

Step 1

There is no surjective homomorphism. Suppose that

$\phi :\mathbb{Z}\left[x\right]\to \mathbb{Z}+\left(3x\right)$.

Since

$deg\phi \left(f\left(x\right)\right)=deg(f\left(\phi \left(x\right)\right)=deg\left(f\left(x\right)\right)deg\left(\phi \left(x\right)\right)$, if $deg\phi \left(x\right)>1$

then nothing in the image could be of degree 1, and the map could not be surjective. So

$\phi \left(x\right)=3ax+b$ for some $a,b\in \mathbb{Z}$

If $\phi$ were surjective, something would have to map to 3x, and this forces $a=\pm 1$

By composing with the isomorphisms of $\mathbb{Z}\left[x\right]$ given by maps $x\mapsto x-c$ and $x\mapsto -x$, we may WLOG assume that $\phi \left(x\right)=3x$

Step 2

The question we ask is, can we find g(x) such that

$\phi \left(g\left(x\right)\right)=g\left(3x\right)=3{x}^{2}$?

Working over $\mathbb{Q}\left[x\right]$, we see if $g\left(3x\right)=3{x}^{2}$, then $g\left(x\right)={x}^{2}/3\overline{)\in}\mathrm{\mathbb{Z}}\left[x\right]$. And because $\phi$ extends to an isomorphism of $\mathbb{Q}\left[x\right]$, this is the only polynomial in $\mathbb{Q}\left[x\right]$ that would satisfy the condition. So the map cannot be surjective.

asked 2020-11-20

Prove that in any group, an element and its inverse have the same order.

asked 2022-05-02

${\mathrm{End}}_{\mathrm{\mathbb{R}}\left[x\right]}\left(M\right)$ where $M=\frac{\mathbb{R}\left[x\right]}{({x}^{2}+1)}$ is a module over the ring $\mathbb{R}\left[x\right]$

asked 2022-05-20

Let a and b belong to a ring R and let m be an integer. Prove that m(ab) = (ma)b = a(mb)

asked 2022-02-28

Compute the kernel for the given homomorphism $\varphi$ .

$\varphi :\mathbb{Z}\to {\mathbb{Z}}_{8}$ such that $phI\left(1\right)=6$ .

asked 2021-09-19

Write formulas for the isometries in terms of a complex variable z = x + iy.

asked 2022-01-12

Let A be a DVR in its field of fractions F, and ${F}^{\prime}\subset F$ a subfield. Then is it true that $A\cap {F}^{\prime}$ is a DVR in ${F}^{\prime}?$ I can see that it is a valuation ring of $F}^{\prime$ , but how do I show, for example, that $A\cap {F}^{\prime}$ is Noetherian?

asked 2021-10-09

If a, b are elements of a ring and m, n ∈ Z, show that (na) (mb) = (mn) (ab)