Let F be a field and let I={an x^n+an-1 x^n-1+⋯+a0 |an, an-1, …, a0 ∈F. and . an+an-

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Let F be a field and let I={an x^n+an-1 x^n-1+⋯+a0 |an,...

Let $F$ be a field and let $$ \begin{aligned} &I=\left\{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \mid a_{n}, a_{n-1}, \ldots, a_{0} \in F\right. \text { and } \\ &\left.\quad a_{n}+a_{n-1}+\cdots+a_{0}=0\right\} \end{aligned} $$ Show that $I$ is an ideal of $F[x]$ and find a generator for $I$.

Let $F$ be a field and let
$$
\begin{aligned}
&I=\left\{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \mid a_{n}, a_{n-1}, \ldots, a_{0} \in F\right. \text { and } \\
&\left.\quad a_{n}+a_{n-1}+\cdots+a_{0}=0\right\}
\end{aligned}
$$
Show that $I$ is an ideal of $F[x]$ and find a generator for $I$.

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Key Concepts

Generators of an Ideal

The concept of a generator of an ideal refers to an element from which every element of the ideal can be obtained by multiplication with any element from the ring. For a principal ideal, the ideal is exactly the set of all multiples of one particular element. This simplifies the study of the ideal's structure and is a fundamental idea in ring theory.

Factor Theorem

The factor theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This theorem is particularly relevant when considering ideals defined by conditions on the coefficients of polynomials, leading to conclusions about divisibility by a specific linear factor.

Ideal in Ring Theory

An ideal in a ring is a subset that is closed under addition and under multiplication by any element of the ring. This concept is central to ring theory and is used to construct quotient rings, which help in understanding the structure of rings by dividing out 'redundant' parts that are captured by the ideal.

Principal Ideal Domain (PID)

A principal ideal domain is an integral domain in which every ideal is generated by a single element. In the context of polynomial rings over a field, such as F[x], the fact that F is a field guarantees that F[x] is a PID, meaning every ideal in F[x] can be written as the set of all multiples of a single polynomial.

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