Intereting Posts

Aronszajn Trees and König's Lemma
How do we show that the function which is its own derivative is exponential?
How many non isomorphic groups of order 30 are there?
What is the general formula for calculating dot and cross products in spherical coordinates?
Homotopy equivalence of universal cover
What are the prerequisites for taking introductory abstract algebra?
Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. Does this imply that $f_n(x)$ is uniform convergent?
Proving: If $|A\times B| = |A\times C|$, then $|B|=|C|$.
proof of the chain rule for calculus
Inner product on $C(\mathbb R)$
Inequalities involving arithmetic, geometric and harmonic means
What is the intuitive meaning of the scalar curvature R?
Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015
$x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$
Find the integral closure of an integral domain in its field of fractions

An exercise from Dummit & Foote:

Determine the units of the ring $A = \mathbb{Z}[X]/(X^{3})$ and the structure of the unit group $A^{\times}$.

Help would be great.

- How to show that $H \cap Z(G) \neq \{e\}$ when $H$ is a normal subgroup of $G$ with $\lvert H\rvert>1$
- Must an ideal generated by an irreducible element be a maximal ideal?
- How to prove this is a field?
- Is a stably free module always free?
- How do I prove $\leq $?
- Characteristic subgroups $\phi(H) \subseteq H$

Thanks!

- Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$
- Variety generated by finite fields
- How to show the Symmetric Group $S_4$ has no elements of order $6$.
- Intuition surrounding units in $R$
- At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic
- Prove that the Gaussian Integer's ring is a Euclidean domain
- In the definition of a group, is stating the set together with the function on it redundant?
- This tower of fields is being ridiculous
- The height of a principal prime ideal
- Double check $G\sim H$ iff $G≈H$

Actually nobody answered the second part of the problem about the structure of the unit group $A^{\times}.$ The answer is the following:

$A^{\times}$ is isomorphic to the abelian group $\mathbb{Z}_2\times\mathbb{Z}^2$.

**Hint.** The unit group can be written as a direct sum of the subgroups generated by $-1$, $1+x$, and $1+x^2$ respectively. The subgroup generated by $-1$ is the torsion subgroup of $A^{\times}$, whilst the subgroup generated by $1+x$ and $1+x^2$ is the free part of $A^{\times}$.

Of course, the problem (and its answer) can be easily generalized to the ring $\mathbb{Z}[X]/(X^n)$.

The key trick is that there is a canonical ring-morphism $A=\mathbb Z[X]/(X^3)=\mathbb Z[x]\to \mathbb Z[X]/(X) \simeq\mathbb Z$ (why?) and that units are sent to units by ring morphisms.

So any unit of $A$ is of the form $u=a+bx+cx^2$ with $a$ a unit in $\mathbb Z$ .

I won’t tell you that $x$ is nilpotent: my colleagues on this site would say that I’m making things too easy for you.

Hints to get you started: every element of $A$ is represented by a unique polynomial $aX^2+bX+c$ with $a,b,c \in \mathbb{Z}$. Make sure you see why. If such an element is a unit, what can you say about $c$? Are there any requirements on $a,b$?

- A (non-artificial) example of a ring without maximal ideals
- Can a number have both a periodic an a non-periodic representation in a non-integer base?
- Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
- Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
- Translations of Kolmogorov Student Olympiads in Probability Theory
- Prove that the rings $End(\mathbb{Z}^{n})$ and $M_{n}(\mathbb{Z})$ are isomorphic
- This sentence is false
- How find this integral $I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\frac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$
- $\sum_{k=-\infty}^\infty \frac{1}{(k+\alpha)^2} = \frac{\pi^2}{\sin^2\pi \alpha}$
- Is $f(z)=\exp (-\frac{1}{z^4})$ holomorphic?
- Why do we need that $\alpha$ is for regular to existence of arc length as integral?
- Compute the derivative of the log of the determinant of A with respect to A
- What is the expectation of $ X^2$ where $ X$ is distributed normally?
- Permutations to satisfy a challenging restriction
- Why are translation invariant operators on $L^2$ multiplier operators