Exercise 1. Given integers d, q, a with $d | q $ and $gcd(a, d) = 1$. Prove that there exists an integer b such that $b \equiv a\ (mod\ d)$ and $gcd(b, q) = 1$.
Answer.
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Exercise 2. Prove that every group of order $p^2$, with $p$ is a prime, is an Abelian group.
Answer.
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Exercise 3. Given an algebraic integer $a$, it means that $a$ is a root of some monic polynomials with coefficients in $\mathbb{Z}$. Prove that the map $f: \mathbb{Z}[x]\rightarrow\mathbb{Z}[a]$ maps $f(x)$ to $f(a)$ with ker is a principal ideal.
Answer.
Not available yet.
Exercise 4. Given an algebraic number $a$, it means that $a$ is a root of some polynomials with coefficients in $\mathbb{Z}$. $R$ is a subring of $Q(a)$ and $I$ is a proper ideal of $R$. Prove that the intersection between $I$ and $\mathbb{Z}$ is always different from $\{0\}$.
Answer.
Not available yet.
Exercise 5. Given G is a finite an Abelian group, $\mathbb{C}$ is the set of complex numbers. Denote $L^2(G)$ is a vector space of functions from G to $\mathbb{C}$, with equipped addition and multiplication: $(f_1+f_2)(g) = f_1(g) + f_2(g)$, $(cf_1)(g) = cf_1(g)$ for all $c \in \mathbb{C}$, $g \in G$ and $f_1, f_2 \in L^2(G)$. Define the inner product for $L^2(G)$: $<f_1,f_2> = \sum_{g\in G} f_1(g)\overline{f_2(g)}$.
1. Prove that $L^2(G)$ is a vector space with dimension $|G|$, and verify the inner product defined above is well-defined.
2. Point out an orthonormal basis of $L^2(G)$.
3. Let $Hom(G, \mathbb{C}^\times)$ is the set of homomorphisms from $G$ to the multiplication group $\mathbb{C}^\times$. Prove that $Hom(G, \mathbb{C}^\times) \cong G$ and the set $Hom(G, \mathbb{C}^\times)$ is an orthogonal basis of $L^2(G)$.
Answer.
Not available yet.
I will update new problems in this blog. I feel these problems are quite hard, if anyone has I solution, please discuss it with me or give me a way to solve. Thank you!
Not available yet.
Exercise 3. Given an algebraic integer $a$, it means that $a$ is a root of some monic polynomials with coefficients in $\mathbb{Z}$. Prove that the map $f: \mathbb{Z}[x]\rightarrow\mathbb{Z}[a]$ maps $f(x)$ to $f(a)$ with ker is a principal ideal.
Answer.
Not available yet.
Exercise 4. Given an algebraic number $a$, it means that $a$ is a root of some polynomials with coefficients in $\mathbb{Z}$. $R$ is a subring of $Q(a)$ and $I$ is a proper ideal of $R$. Prove that the intersection between $I$ and $\mathbb{Z}$ is always different from $\{0\}$.
Answer.
Not available yet.
Exercise 5. Given G is a finite an Abelian group, $\mathbb{C}$ is the set of complex numbers. Denote $L^2(G)$ is a vector space of functions from G to $\mathbb{C}$, with equipped addition and multiplication: $(f_1+f_2)(g) = f_1(g) + f_2(g)$, $(cf_1)(g) = cf_1(g)$ for all $c \in \mathbb{C}$, $g \in G$ and $f_1, f_2 \in L^2(G)$. Define the inner product for $L^2(G)$: $<f_1,f_2> = \sum_{g\in G} f_1(g)\overline{f_2(g)}$.
1. Prove that $L^2(G)$ is a vector space with dimension $|G|$, and verify the inner product defined above is well-defined.
2. Point out an orthonormal basis of $L^2(G)$.
3. Let $Hom(G, \mathbb{C}^\times)$ is the set of homomorphisms from $G$ to the multiplication group $\mathbb{C}^\times$. Prove that $Hom(G, \mathbb{C}^\times) \cong G$ and the set $Hom(G, \mathbb{C}^\times)$ is an orthogonal basis of $L^2(G)$.
Answer.
Not available yet.
I will update new problems in this blog. I feel these problems are quite hard, if anyone has I solution, please discuss it with me or give me a way to solve. Thank you!
Have no idea about the first one but the second one is quite easy. We can have a better statement than that by classifying the group. If group G has order p^2 then it is isomorphic to $Z_p\times Z_p$ or $Z_{p^2}$. Both cases, G is abelian. Now how to classify G. By look into the order of elements of G, if there is an element of order $p^2$ hence G is cyclic of order $p^2$. Otherwise, every non-identity elements of $G$ has order $p$ hence G is isomorphic to $Z_p\times Z_p$.
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DeleteThanks for your proof, Mr. Tuan. I will read and try to understand what you wrote. After that, I will update the solution.
DeleteI have updated new exercises in this blog, hope you enjoy it.
Regards,
Tang Khai Hanh
The last sentence of Mr. Tuan can be understood by choosing an element $a$ of order $p$, then consider the cyclic group generated by $a$. Pick any element outside this, namely $b$. Then we next prove that the two cyclic groups generated by $a$ and $b$ intersects only at the identity element. It gives the explicit group structure.
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