# Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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### Prove linear independence and spans with linear maps

Suppose that $V,W$ are vector spaces over $\Bbb{F}$ and that $T : V → W$ is a linear transformation. (a) Suppose that $T$ is one-to-one, and that $\{v_1, · · · , v_n\}$ is linearly independent in $V$ ...
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### Show that the alternating group $A_9$ has no subgroups of index 8?

So far, I believe it's a proof by contradiction. Suppose that $H \leq A_9$ with $[A_9 : H] = 8$.. $|H| = |A_9|*8$(which is a large number)? then would this involve the 3-cycles? Quite stumped. Thank ...
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### Isomorphism of direct product of quotient rings

How do I show that with $n = p^aq^b$ with $p,q$ distinct primes and $a,b \geq 1$ that $$\mathbb Z/n\mathbb Z \cong (\mathbb Z/p^a\mathbb Z) \times(\mathbb Z/q^b\mathbb Z)?$$ I am told that Bezout's ...
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### Is every well ordered commutative nontrivial ring with identity an well ordered integral domain?

$\mathbb Z$ is up to ring isomorphism the only well ordered domain, that is, $\mathbb Z$ is a integral domain and every nonempty subset of $\{n \in \mathbb Z: n\geq 0\}$ has a least element. But what ...
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### automorphism group of direct product of groups

I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups. Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then ...
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### Sylow counting - classifying groups of order 15

let $G$ be a group of order $15$. this is the argument I was given: We have $15 = 3\times 5$ so we start with $p = 5$ We have by Sylow's theorem that $N_5 = 1 \mod 5$ so $N_5 = 1$ or $N_5 \geq 6$. ...
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### Galois theory about automorphisms of the field of rational functions

Suppose That $F$ is a field and $G=Aut(F(x))$ is the group of field automorphisms of the field of rational functions $F(x)$ and fix $F$, and that $E\subset F(x)$ is the fixed field of G. please prove ...
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### Show that $a(-1) = (-1)a = -a$.

In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
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### Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
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### For $\mathbb{Q[x]}/ I \cong$ $\mathbb{Q}$, proving kernel

Let I $=<x-2>$. Prove $\mathbb{Q[x]}/I \cong \mathbb{Q}$ $\textbf{Pf:}$ Define $\phi: \mathbb{Q[x]} \rightarrow \mathbb{Q}$ by $\phi(f(x)) = f(2)$ I understand how to show that it is ...
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### What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
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### Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
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### When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?

Let $x\in\mathbb{C}$, and consider the field $\mathbb{Q}(x)$. If this field is a finite extension of $\mathbb{Q}$, then $x$ is algebraic over $\mathbb{Q}$, so satisfies some polynomial $P$. Any field ...
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### the cardinal of the set of left cosets the same as the cardinal of the set of right cosets?

is $|G/H| = |H/G|$ where $G/H$ is the set of left cosets of H in G, and $H/G$ the set of right cosets of H in G? I know that $|gH| = |H| = |Hg|$ but I don't see how $|G/H| = |H/G|$, even though ...
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### Is $\mathbb{Q}(\sqrt{2},\sqrt[3]{5})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$?

I understand that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ But I am struggling to algebraically show that $\sqrt{2}$,$\sqrt[3]{5}\in\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$...
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### Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus \mathbb{Z}$,...
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### Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
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### Examples for Burnside problem.

What are some examples for Burnside Problem- example of an infinite finitely generated torsion group - except Grigorchuk group. I have studies Grigorchuk group as an counterexample which was first ...
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### Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
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### About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
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### Question about the centralizer and conjugacy classes

Let $G$ be a finite $p$-group and $H$ a non trivial normal subgroup of $G$. I want to prove that $H\cap Z(G)\neq 1$. I define a relation in $H$ by $x\sim y$ if and only if there exists $g\in G$ such ...
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### Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3).$

How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of ...
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### Why $K = (X_1, X_2, …)$, the ideal generated by $X_1, X_2, …$ not finitely generated as a R-module?

Let $R = \mathbb{Z}[X_1, X_2, \dots]$ be the ring of polynomials in countably many variables over $\mathbb Z$. Why $K = (X_1, X_2, ...)$, the ideal generated by $X_1, X_2, ...$ is not finitely ...
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### is 0 in the following Ideal?

Given $R=\mathbb R[x]$ and $I=(2x^3-3x^2+2x-3)+(2x^2-x-3)$ Is an Ideal of R? I don't understand what the quantity I is... Am I supposed to sum them together giving $2x^3-x^2+x-6$ Now here's the ...
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### Quadratic number field which is Euclidean but not norm Euclidean [closed]

I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2$ or $3\pmod 4$ , whose ring of integers is Euclidean but not norm Euclidean. Please help. Thanks in advance....
Say if we have $I+a+bX+cX^2+I+I$, can we rearrange the order to how we like? Because you can always imagine $0+I$ when the ideals are written consecutively.
If $F$ is a field and $p(x) \in F[x]$, prove that the ring $R=F[x]/(p(x))$ has no nonzero nilpotent elements iff $p(x)$ is not divisible by the square of any polynomial. (==>) $R$ has no nonzero ...