# explicit generators of a quotient polynomial ring associated to a given polynomial

Let $k$ be a field and consider ring of polynomials $k[X]$. If $F(X)\in k[X]$ and $W(X)\in k[X]$, then we will write $\overline{W(X)}$ to denote element in $k[X]/(F(X))$ represented by $W(X)$.

Lemma. Assume, that ${a}_{1},\mathrm{\dots},{a}_{n}\in k$ are different elements and $F(X)=(X-{a}_{1})\mathrm{\cdots}(X-{a}_{n})$. Let ${W}_{i}(X)\in k[X]$ be given by ${W}_{i}(X)=(X-{a}_{1})\mathrm{\cdots}(X-{a}_{i-1})\cdot (X-{a}_{i+i})\mathrm{\cdots}(X-{a}_{n})$. Then there exist ${\lambda}_{1},\mathrm{\dots},{\lambda}_{n}\in k$ such that $F(X)$ divides polynomial^{}

$$U(X)=\left(\sum _{i=1}^{n}{\lambda}_{i}\cdot {W}_{i}(X)\right)-1.$$ |

Proof. Note, that ${W}_{i}({a}_{i})\ne 0$ for any $i$. Thus we may define ${\lambda}_{i}={\left({W}_{i}({a}_{i})\right)}^{-1}$. Then for any $i$ we have ${\lambda}_{i}\cdot {W}_{i}({a}_{i})=1$, therefore

$$V(X)=\sum _{i=1}^{n}{\lambda}_{i}\cdot {W}_{i}(X)$$ |

is such that $V({a}_{i})=1$ for any $i$. In particular $U({a}_{i})=V({a}_{i})-1=0$ and thus $(X-{a}_{i})$ divides $U(X)$ for any $i$. This completes^{} the proof. $\mathrm{\square}$

Corollary. Under the same assumptions^{} as in lemma, we have that ideal $(\overline{{W}_{1}(X)},\mathrm{\dots},\overline{{W}_{n}(X)})$ in $k[X]/(F(X))$ is equal to $k[X]/(F(X))$.

Proof. Indeed, all we need to show is that we can generate $\overline{1}$. Lemma implies, that there is $V(X)\in k[X]$ such that

$$\left(\sum _{i=1}^{n}{\lambda}_{i}\cdot {W}_{i}(X)\right)-1=F(X)\cdot V(X).$$ |

Now, after aplying quotient^{} homomorphism^{} to both sides we have

$$\sum _{i=1}^{n}{\lambda}_{i}\cdot \overline{{W}_{i}(X)}=\overline{1}.$$ |

This completes the proof. $\mathrm{\square}$

Remark. This gives us an explicit formula for generators^{} of $k[X]/(F(X))$. In particular the dimension over $k$ of this ring is at most $\mathrm{deg}F(X)$. It can be shown that actualy it is equal, even if $F(X)$ is arbitrary.

Title | explicit generators of a quotient polynomial ring associated to a given polynomial |
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Canonical name | ExplicitGeneratorsOfAQuotientPolynomialRingAssociatedToAGivenPolynomial |

Date of creation | 2013-03-22 19:10:01 |

Last modified on | 2013-03-22 19:10:01 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 6 |

Author | joking (16130) |

Entry type | Derivation |

Classification | msc 11C08 |

Classification | msc 12E05 |

Classification | msc 13P05 |