This exam is 2 hours long.
Your numerical answers should be EITHER exact OR correct to 3 decimal places.
For each question, you must show all of your working steps in order to receive full points. Zero point will be given if you only provide an answer without any working steps.
Cheating is a serious offense. Students caught cheating are subject to a zero score as well as additional penalties.
Question 1
Determine, with justification, wether the following ideals are prime, maximal, both, or neither.
(a) $\left\langle x^{3}-1\right\rangle$ in $\mathbb{Q}[x]$.
(b) $\left\langle x^{9}+7\right\rangle$ in $\mathbb{Q}[x]$
(c) $3 \mathbb{Z} \times 5 \mathbb{Z}$ in $\mathbb{Z} \times \mathbb{Z}$
(d) $\langle 2 x\rangle$ in $\mathbb{Z}[x]$.
(e) $\langle x\rangle$ in $\mathbb{Z}[x]$
用素理想和极大理想的定义验证即可
Quesulon 2
Consider $x^{2}+1$ and $x^{2}-1$ in $\mathbb{Z}{7}[x]$. (a) Show that $x^{2}+1$ is irreducible and that $x^{2}-1$ is not irreducible. (b) Show that both $\mathbb{Z}{7}[x] /\left\langle x^{2}+1\right\rangle$ and $\mathbb{Z}{7}[x] /\left\langle x^{2}-1\right\rangle$ have 49 elements. (c) Show that $\mathbb{Z}{7}[x] /\left\langle x^{2}+1\right\rangle$ is a field, but $\mathbb{Z}_{7}[x] /\left\langle x^{2}-1\right\rangle$ has zero divisors.
用爱森斯坦判别法
Question 3
Use less than 30 words to answer each of the following questions:
Let $F$ be a field and let $a, b \in F$ with $a \neq 0 .$ Show that
$$
F[x] /\langle a x+b\rangle \simeq F
$$
把自然的同构映射构造出来并证明这是一个同构
Question 4
Consider the map
$\varphi: \mathbb{Z}{7}[x] \rightarrow \mathbb{Z}{7}[x] /\langle x+1\rangle \times \mathbb{Z}{7}[x] /\langle x-1\rangle$
defined by $\varphi(f(x))=(f(x)+\langle x+1\rangle, f(x)+\langle x-1\rangle)$.
(a) Show that $\varphi$ is a homomorphism of rings.
(b) Show that $\operatorname{ker}(\varphi)=\left\langle x^{2}-1\right\rangle$.
(c) show that $$ \mathbb{Z}{7}[x] /\langle x+1\rangle \times \mathbb{Z}{7}[x] /\langle x-1\rangle \simeq \mathbb{Z}{7} \times \mathbb{Z}{7} $$
(d) Use these and problem $3(\mathrm{~b})$ to show that $$ \mathbb{Z}{7}[x] /\left\langle x^{2}-1\right\rangle \simeq \mathbb{Z}{7} \times \mathbb{Z}{7}
$$
按照定义验证
Question 5
Read the data from the file exam_data.txt.
a) [3 pts] Find the sample mean of the daily-confirmed cases for each week.
b) Suppose the daily-confirmed cases for each week follow normal distributions with equal variances. They are also
Let $I_{1}$ and $I_{2}$ be two ideals in a ring $R$. Consider
$$
\varphi: R \rightarrow R / I_{1} \times R / I_{2}
$$
defined by $\varphi(r)=\left(r+I_{1}, r+I_{2}\right)$
(a) Show that $\varphi$ is a homomorphism of rings.
(b) Find the kernel of $\varphi$.
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