Intereting Posts

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Prove that $f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(Re^{i\phi})Re(\frac{Re^{i\phi}+z}{Re^{i\phi}-z}) d\phi$
Is there a name for a group having a normal subgroup for every divisor of the order?
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If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

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.

- Ideal of polynomials in $k$ vanishing at a point $p$ is $(X_1 - p_1, …,X_n - p_n)$
- Does validity of Bezout identity in integral domain implies the domain is PID?
- Is there a proper subfield $K\subset \mathbb R$ such that $$ is finite?
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- Solve $f^n(x)=4^nx+\frac{4^n-1}{3}$ for $n$.
- Is every regular element of a ring invertible?

Thanks!

- Proving equivalence relations
- How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?
- Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
- Show that $G\to\operatorname{Aut}(G)$, $g\mapsto (x\mapsto gxg^{-1})$ is a homomorphism
- Prove that $(3, 2 + \sqrt{-5})$ is maximal ideal in $\mathbb Z$
- Non-abelian group of order $p^3$ without semidirect products
- Sum of nilpotent and element is a unit in ring?
- Irreducible but not prime in $\mathbb{Z} $
- Embedding Fields in Matrix Rings
- Explaining the product of two ideals

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 $a,b$?

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