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RSS Abstract Algebra on Math.StachExchange.com

  • Prove that if $h(x)$ divides $g(x)$ in $\mathbb{Q}[x]$, then $h(x)$ divides $g(x)$ in $\mathbb{Z}[x]$ [on hold]
    Let $g(x)$ and $h(x)$ belong to $\mathbb{Z}[x]$ and let $h(x)$ be monic. If $h(x)$ divides $g(x)$ in $\mathbb{Q}[x]$ show that $h(x)$ divides $g(x)$ in $\mathbb{Z}[x]$.
  • Must $G$ be a transitive group to ensure bijection between orbits?
    In Godsil and Royle's Algebraic Graph Theory, they prove this lemma about orbits of a transitive group action: Let $G$ be a transitive group acting on a set $V$, and let $x$ be a fixed element of $V$. Then there is a bijection between orbits of $G$ on $V\times V$ and orbits of $G_x$ on $V$. Where in the following proof is the assumption that $G$ is tran […]
  • Herstein Problem No.13 Page 109
    Give an example of a finite non-abelian group $G$ which contains a subgroup $H_0 \neq (e)$ such that $H_0 \subset H$ for all subgroups $H \neq (e)$ of $G$. Can someone help me please?
  • Efficient way to find $[\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{2}): \mathbb{Q}(\sqrt{3}, \sqrt{2})]$ [duplicate]
    This question already has an answer here: Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$ 3 answers I want to show rigorously that this is 2. I'm sure there's a faster way than by trying to see if $\sqrt{5} = a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$ (linear combination of the basis elements for $\mathbb{Q}(\sqrt{3}, \sq […]
  • order of quotient ring
    In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$ In fact, $I=\left \langle x,2 \right \rangle$ How do I show that the order of $\mathbb{Z}\left [ x \right ]$/I is 2?
  • Equivalent definitions of separable polynomial
    Let $T\leq U \leq K$ be fields, with $K$ being an algebraic closure of $T$. We will define the degree of separation of $U$ over $T$, denoted by $[U:T]_s$, as the cardinality of the set $\{ \varphi | \varphi:U \rightarrow K,\varphi$ is a $T$-isomorphism $\}$. It can be checked that this does not depend on the choice of $K$, and thus the definition is correct. […]
  • $F(u) = F(u^2)$ if $u$ is algebraic of odd degree
    This has been asked before, but I have a different solution and I would like to check it. Let $F$ be a field and $u$ be algebraic over $F$ of odd degree. Prove that $F(u) = F(u^2)$. Of course $F(u^2) \subset F(u)$. If we can show that $u \in F(u^2)$, we are through. Let $$f(x) = x^{2n+1} + a_{2n}x^{2n} + \cdots + a_1 x + a_0$$ be the minimum polynomial of $u […]
  • If two sheaves have same stalks everywhere, then two sheaves are isomorphic?
    This is a problem of Tennison Sheaf Theory Problem 2.1 "Let $I=[0,1]\subset R$, Show there is a unique(up to isomorphism) sheaf of abelian groups $F$ on $I$ with stalks:$F_0=F_1=Z, F_x=\{0\}$ for $x\in I-\{0,1\}$" This is basically constructed from patching two skyscraper sheaves. Consider $F(U)=Z$, for either $0$ or $1\in U$, and $F(U)=0$, for any […]
  • Normal Group and Conjugacy Class
    I am try to solve the following: Let $G$ a group and $N$ a normal subgroup of $G$ with index a prime $p$. Consider $C$ a conjugacy class of $G$, which is contained in $N$. Prove that either $C$ is a conjugacy class of $N$ or it is a union of $p$ distinct conjugacy classes of $N$. The first idea I had was using some group actions properties, but I couldn […]
  • About the irreducibility and separability of a polynomial
    Theorem: If the extension $E/F$ is finite and Galois, then $E/F$ is normal and separable. Proof: Let $f(x)\in F[x]$ irreducible with a root $\alpha\in E $ and $\sigma_1(\alpha),\dots,\sigma_r(\alpha)$ the orbit of $\alpha$ under $G(E/F).$ The polynomial $\prod_{i=1}^{r}(x-\sigma_i(\alpha))$ is separable and irreducible in $F[x].$ Therefore $\prod_{i=1}^{r}(x […]

Final Grades

Your grades were submitted on Titan Connect.

Most likely they will not be “rolled-in” until Monday.

I wish you have a good break and all the best for your next semester.

Last minute questions

If you have any last minute questions, I will be in my office from 1 to 2 today.

Last Homework Assignment- Due Monday 12/8

Section 6.8,

Problem 4 (see definitions in section 6.1), 9(a), 10(a),

Section 6.11

Problems 2, 3, 7(a)-(d)

Section 6.12

Problems 1, 3, 14

Section 6.13

Problems 5, 8, 9,

Section 13.2

Problem 6

Section 13.3

Problems 4, 5

Presentation Problem for Monday 12/8

Section 13.2, Problem 4.

Presentation Problem for Wednesday 12/3

Problem based on Section 6.13, Problem 3:

Let G be a group and for each x in G, let T_x be the inner automorphism T_x(a)=x^(-1) a x.

Define a function phi: G->Inner automorphisms of G by phi(x)=T_x.

Prove that the kernel of phi is the center of G (see definition in pg. 144) , and that phi is onto.

Then use the First Isomorphism Theorem to prove that G/Z is isomorphic to the inner automorphisms of G, where Z is the center of G.

Course Evaluations

All students are encouraged to complete their course evaluations. Go to http://www.udmercy.edu/evaluate
Log in with your TitanConnect login credentials.
Complete the course evaluations.
Thank you.

Presentation Problem for Monday 12/1

Section 6.12,
Problem 2.