Polynomial ring, ideals and Spec

Question

Morning everyone, I want some hint about this.

i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set

ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain

iii)Determine its Spectrum(Spec)

My Answers

i) ideals are the form of $\frac{P}{<X^3-1>}$ where $P$ contains $<X^3-1>$ but I don't know how to define P

Answer 1

For i), it means $P$ is divisible by $X^3-1$. In other words: $P(1)=0\;$ and $\;P(\mathrm e^{\tfrac{\mathrm i\pi}3})=0$.

ii) Hint: use the Chinese remainder theorem.

iii) Hint: think of the factorisation of $X^3-1$ into irreducible factors and use i).

Answer 2

the ring $\Bbb{R}[X]$ is principal ring because $\Bbb{R}$ is a field. the ideal in this quotient are in the forme classes of $\langle P(X)\rangle =\langle P(X)\rangle +\langle X^3-1\rangle =\{S(X)P(X)+T(X)(X^3-1), S,T\in \Bbb{R}[X]\}$ and better we can represent this class by class of $\langle R(X)\rangle$ where $R(X)$ is the rest by euclide division of $P(X)$ by $X^3-1$.

the polynomial $X^3-1$ is not prime because not irreducible so the ideal $\langle X^3-1\rangle$ not prime and therefore the quotient ring is not integre.

Let $J$ an ideal in the quotient the canonical projection asure that $J$ is a class of $\langle P(X) \rangle$ where $ \langle X^3-1\rangle \in \langle P(X)\rangle$ if we loock , at the compositum of canonical projection $ \Bbb{R}[X]\twoheadrightarrow \Bbb{R}[X]/ \langle X^3-1\rangle\twoheadrightarrow \Bbb{R}[X]/ \langle X^3-1\rangle/J$, by isomorphisme theorem, we can say that J is prime in $\Bbb{R}[X]/ \langle X^3-1\rangle$ if and if $P$ is a irreducible factor of $X^3-1$ that is $P=x-1$ or $ X^2+X+1$.