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 that $I[x]$ is not maximal in $R[x]$.

I've asked a lot tonight, but I'm grinding through my HW assignment and it's kicking my butt. This is my final problem, and I've managed to only struggle on 3 of the 13 problems, which I think is ...
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The mapping $\theta : S^{-1}R \rightarrow (\pi(S))^{-1}(R/I)$ is a well-defined ring epimorphism.

I'm working on this problem for a homework assignment. Note that $R$ is a commutative ring with unity, $I$ is an ideal of $R$, and $\pi : R \to R/I$ is the canonical projection given by $\pi(r)=r+I$. ...
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Sets of non-complements elements in a lattice.

Let $L$ be a finite lattice with a least element $0$ and a greatest element $1$, where $0\neq 1$. Fix a $t\in L$, and let $X$ be the set of non-complements of $t$, i.e., the set of all $x$ such that ...
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Rentschler's theorem on non-commutative algebras

Rentschler's theorem says that every locally nilpotent derivation of the algebra $A=\mathbb{C}[x,y]$ (i.e., a linear map $\phi$ that satisfies the Leibniz rule and such that for every $p \in A$ ...
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Show that if $F$ is a field, then $<x>$ is maximal in $F[x]$. Also, show that $F[x]$ is not local.

See statement above. So far I have the following: Assume that $<x>$ is not maximal. Then $ <x> \subset <f(x)> \neq F[x]$. This means that $x = f(x) g(x)$. Since $x$ is ...
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1answer
60 views

Prerequisite books before Hungerford's Algebra?

Prerequisite books before Hungerford's graduate Algebra? I have an pdf version of the book and feel the Hungerford is overcomplicated after i finish some of the books title with something like first ...
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60 views

Field $K (x)$ of rational functions with coefficients from $K$, if $f\in K(x)$, then $f^2 \neq x^2-1$

I'm in the process of studying for an exam and I came across the following question: Prove that if $K$ is a field and $K (x)$ is the field of rational functions with coefficients from $K$, if ...
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Prove that $x^2 + 3x +2$ is irreducible in $\mathbb{Z}[[x]]$, but not in $\mathbb{Z}[x]$.

As the problem states, I need to show irreducibility of the given polynomial. I'm not sure where go with this, so any help would be great. I know that Eisenstein has a nice test for this in ...
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3answers
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Prove the set $\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$ is a ring.

Prove that if $d$ is a non-square integer with $d \equiv 1 \mod 4$ then the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$$ is a ring, and in ...
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Proper subgroup of a $p$-group.

Let $G$ be a $p$-group and $H$ a proper subgroup such that $|H|=p^s$. I want to prove that there exists $H'$ subgroup of $G$ such that $H\subset H'$ and $|H'|=p^{s+1}$. I don't know if induction ...
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Counting subgroups of a $p$-group.

Let $G$ be a finite $p$-group, say $|G|=p^n$, and let $0\le k\le n$. Call $\mathcal{A}$ the set of subgroups of order $p^k$, and $\mathcal{N}\subseteq\mathcal{A}$ the subgroups that are normal. I ...
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2answers
39 views

Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. [duplicate]

Suppose $m, n > 1$ are positive integers which are relatively prime. Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. Two of them are $[0], [1]$, how will I find the other two?
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Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
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20 views

Must factors of a monic polynomial over an integral domain be monic? [on hold]

Must factors of a reducible monic polynomial over an integral domain be monic? If so, how can I prove it? If not, what counterexample is there?
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How's this inertia called?

Let $E/F$ be an algebraic extension. Let $L_1,L_2$ be algebraically closed fields and $\sigma_1:F\rightarrow L_1,\sigma_2:F\rightarrow L_2$ be field monomorphisms. Define ...
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51 views

Second Isomorphism Theorem

There is one little detail in the proof I would very much like to get your opinion of. Look at where I have circled in red: There it seems that they have used that $\mu_2((hn)N)=h$. But isn't ...
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30 views

Normal subgroup corresponding to a relation

Suppose I have a free group on $n$ elements, $FX$, quotient-ed by an element (say, $\langle a, b \rangle/aba^{-1}b^{-1}$), how do I compute the normal subgroup $N$ of $FX$, such that $FX/N$ matches ...
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2answers
24 views

Does every algebraically closed field with nonzero characteristic have a unique finite subfield $p^n$?

Let $F$ be an algebraically closed field such that $char(F)\neq 0$. Then, $\forall n\in\mathbb{Z}^+$, there exists a unique finite subfield $K$of $F$ such that $|K|=char(F)^n$. Is this ...
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Must unital ring homomorphism be identity mapping?

Unital ring homomorphism is one that maps 1 to 1. So I reason it this way, $\phi(a) = \phi(\overbrace{1+...+1}^a)= \overbrace{\phi(1) + ... + \phi(1)}^a = \overbrace{1 + ... + 1}^a = a.$ Hence, it is ...
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Difference between generators and basis

What is the difference between the terms "generator set" and "basis"? Don't they both just mean a set of objects that you can use to obtain all of the objects in a larger set under some operations? ...
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The zeroes in $\mathbb{C}$ of $x^{28}-1 ∈\mathbb Q[x]$ form cyclic group under multiplication [on hold]

The zeroes in $\mathbb{C}$ of $x^{28}-1 ∈\mathbb Q[x]$ form cyclic group under multiplication, explain this.
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Invertible matrix that sets some rows or columns of product to 0 or 1

Assume that there is a fixed binary matrix denoted by $B$ that is a $n\times n$ matrix. " I need to know whether exists a binary matrix $A$ that is both invertible and make matrix $C$ to have some ...
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59 views

Which group of order 16 is this? [on hold]

$G$ is an abelian group of order $16$ that has elements $a$ and $b$ such that $|a| = |b| = 4$ and $a^2$ does not equal $b^2$. What group is $G$ isomorphic to?
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35 views

Proving that the Galois group of minimal polynomial of constructible number is a power of $2$

I was trying to understand whether a real number has degree which is power of 2 over rationals is constructible. Then I found this. But I am having trouble in understanding why this statement has to ...
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1answer
51 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...
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Show that $Z(θ^G)≤H$

Suppose $H ≤ G$ and $θ \in Char(H)$. Show that $Z(θ^G)≤H$. ($Z$ is the centre and $θ^G$ is the character induced by $G$)
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Prove basis exists for a weird transformation

Suppose that $S : \Bbb{R}^2 → \Bbb{R}^2$ is a linear transformation such that $S^2 = S, S\ne 0 $ and $ S \ne I$ Prove that there is a basis for $\Bbb{R}^2$ with respect to which the matrix $A_S$ of ...
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$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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Prove that a non empty finite subset H of a group G that shows closure is a subgroup.

This might be a little trivial but I am having difficulty in proving H will always consist of an identity element. Thanks
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27 views

Define Tame algebra

How would you define a tame algebra such that undergraduate students could understand it easily? Do someone knows a good book that has a nice way to explain it?
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Constructing the groupification of a semigroup (Vakil 1.5.G)?

In the first chapter of Ravi Vakil's Algebraic Geometry notes, he suggests a construction of the groupification $H(S)$ of an abelian semigroup $S$ by considering $S\times S/\sim$ where $(a,b)\sim ...
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I have to show Two monic polynomial of the same degree [on hold]

I have to show Two monic polynomial of the same degree coincide if and only if they have the same zero set over algebraically closed field
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0answers
33 views

Determining Quotient group

Let $G$ be the group of linear functions under addition. Let $H$ be the subgroup of $G$ containing only the linear functions passing through the origin. How do I determine the group $G/H.$ What are ...
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1answer
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Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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size of centralizer

Can someone tell me why the size of the centralizer of the semidirect product of $\mathbb{Z}_4\times \mathbb{Z}_4$ and $S_2$ is $4^2 \times 2 = 32$? I know the part why $4^2$ is multiplied by 2 that ...
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2answers
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Checking for the reducibility of a polynomial using rational root theorem

When checking for the reducibility of a polynomial over $\mathbb{Z}$. I can either use the eisenstein criteria or contradiction. However, I am wondering if it is possible to use rational root theorem. ...
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1answer
26 views

Proof of that in an integral domain, every prime element is irreducible.

I would like to prove that in an integral domain $R$, every prime element $p$ is irreducible. I understand the case where $p = ab$ but the textbooks I have read do not address the case where $p \neq ...
3
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1answer
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let $G=\mathbb{Q}^*$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization. Prove it is a group isomorphism

let $(G, \cdot)=(\mathbb{Q}^x, \cdot) = (\{\frac{p}{q}\mid\frac{p}{q} \neq 0\}, \cdot)$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization and $\varphi$ is ...
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16 views

Proving existence of Hermitian Adjoint in unusual way

For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$. There are two things I'm required to ...
2
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1answer
40 views

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$.

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$. We consider an ideal $J$ such that $I \subset J\subset\Bbb Z[i] $. So there exists an element $p \in J$ but ...
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Prove vector spaces are isomorphisms

Suppose that $V$ is a finite-dimensional vector space over $\Bbb{F}$ and that $T : V → V$ is an isomorphism. Prove that if $S : V → V$ is also a linear transformation and $ST$ is an isomorphism, then ...
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Suppose $E$ is the quotient field of $D$ then find the relation between $D[x]$ and $E[x]$.

Let $D$ be an integral domain, then $D[x]$ is an integral domain and find its quotient field. Suppose $E$ is the quotient field of $D$. Then find the relation between $D[x]$ and $E[x]$. I have ...
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1answer
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Let N be a normal subgroup of a group G. Prove that G is a p-subgroup if both N and G/N are p-subgroups. [on hold]

Could someone please hint the outline of the proof of this Sylow p-theorem question? If N and G/N are p-subgroups, how do I "connect" this to G? Thanks in advance.
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1answer
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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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1answer
32 views

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 ...
3
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1answer
53 views

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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1answer
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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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Proof theorem of Lie's Algebra [on hold]

I need help with a proof this theorems, anyone could be help? I need a proof this theorems. Corollary of Lie's Theorem: Let $L$ be a solvable subalgebra of $\text{gl}(V)$, $\dim V = n$ (finite). ...
2
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0answers
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Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...