LEC ALG3 11

[!Remark]
Let $R$ be a Noetherian ring, and let ${\mathfrak{m}}_1,\dots ,{\mathfrak{m}}_k$ be maximal ideals. Then ${\mathfrak{m}}_1^{a_1}\dots {\mathfrak{m}}_k^{a_k}={\mathfrak{m}}_1^{a_1}\cap \cdots \cap {\mathfrak{m}}_k^{a_k}$.

If $I$ is any ideal and $I={\bigcap }_{i=1}^k{\mathfrak{m}}_i^{a_i}$ where ${\mathfrak{m}}_i$ are maximal. Let $S=R/I$. Then the prime ideals in $S$ are $\{({\mathfrak{m}}_i+I)/I|i\}$. In fact, all of these are maximal. These are called Artinian rings.

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Gauss primes

see Butler (n.d.)

Example 116.1(Example).

Let $\alpha$ be the $p$th root of unity, where $p$ is a prime. Then,

$$\mathbb{Z}[\alpha ]\cong \frac{\mathbb{Z}[x]}{1+x+\dots x^{p-1}}.$$

Let $f(x)=1+x+\cdots +x^{p-1}$. Clearly, $f(\alpha )=0$. Use https://en.wikipedia.org/wiki/Eisenstein%27s_criterion to prove $f$ is irreducible.

Example 116.2(Example).

To show that $3$ is prime in $\mathbb{Z}[i]$, it is sufficient to show that

$$\frac{\mathbb{Z}[i]}{(3)}\cong \frac{{\mathbb{F}}_3[x]}{x^2+1}$$

is an integral domain. $x^2+1\in {\mathbb{F}}_3[x]$ is prime iff it is irreducible, and it is reducible iff it can be factored into monic factors, in which case it must have a root. It can be easily checked that $\overline{0},\overline{1},\overline{2}$ are not roots of $x^2+1$.

On the other hand, $x^2+1$ IS reducible in ${\mathbb{F}}_5[x]$, so $5$ is not prime in $\mathbb{Z}[i]$.

If $I$ is an ideal in a ring $R$, then $\frac{R}{I}[x]\cong \frac{R[x]}{IR[x]}$.

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[!Exercise]
Let $p$ be a prime number and $n=p+1$. Let $f(x)=x^{2p}+nx+n\in Q[\sqrt[n]{p}][x]$. Is $f(x)$ irreducible?

[!Lemma]
$d|a+bi$ iff $d|a$ and $d|b$.

[!Lemma]
Let $\pi$ be a Gauss prime. Then $\pi \overline{\pi }$ is either a prime integer or the square of a prime integer.

[!Proof]-
$\pi \overline{\pi }\in \mathbb{Z}$.

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References

Butler, L. A. (n.d.). A CLASSIFICATION OF GAUSSIAN PRIMES. https://people.maths.bris.ac.uk/~malab/PDFs/2ndYearEssay.pdf