31 questions linked to/from Characterizing units in polynomial rings
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### Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$ [duplicate]

Is there a method to find the units in $\mathbb{Z}_{n}[x]$? For instance, take $\mathbb{Z}_{4}$. How do we find all invertible polynomials in $\mathbb{Z}_{4}[x]$? Clearly $2x+1$ is one. What about ...
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### What are the units of Z[x]? [duplicate]

Where $\mathbb{Z}[x]$ is the ring of polynomials in $x$ with integer coefficients. The book I am studying says the unity of this ring is $f(x) = 1$ so then if some $p \in \mathbb{Z}[x]$ is a unit, ...
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### Units in a polynomial ring [duplicate]

I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the unit group of a ring $R$. I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are ...
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### units of polynomial rings [duplicate]

When does a polynomial in the ring of polynomial have an inverse? I thought only constant polynomials were units. if there are other units, under what rings can we guarantee the existence of inverse ...
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### Which are the units of $R[x]$? [duplicate]

In general be $R$ a ring. So what are $u(R[x])$? where $R[x]$ denotes the rings of polynomials with coefficients in $R$. For example $u(\mathbb{Z}[x]) = \{-1,1\}$
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### Why is 1 not a prime number?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
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### $x$ not nilpotent implies that there is a prime ideal not containing $x$.

Let $\mathscr{N}(R)$ denote the set of all nilpotent elements in a ring $R$. I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal ...
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### Units of polynomial rings over a field

If $R$ is a field, what are the units of $R[X]$? My attempt: Let $f,g \in R[X]$ and $f(X)g(X)=1$. Then the only solution for the equation is both $f,g \in {R}$. So $U(R[X])=R$, exclude zero elements ...
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### Under what conditions is a zero divisor element $a$ in commutative ring $R$ nilpotent?

Suppose that $R$ is a commutative ring with identity $1$ Let $a\in R$ with $ab=0$ for some $b\ne0$. Under what conditions $a$ must be also nilpotent?
### Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit.
Q: Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit. I know (2x+1) is a unit. Is there any other units in $Z_4[X]$ if there are infinite units, could you generalize it in a certain ...