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

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Abstract Algebra: Prove that every field has only trivial ideals

Prove that every field has only trivial ideals (that is, {0} and the field itself)
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Prove something is divisible by a prime [on hold]

Let $p$ be a prime. Prove that $\sum_{k=j}^{p-1} \frac{k!}{ j! (k-j)! }$ is divisible by $p$ $\forall$ $j \in \{0, ..., p-2\}$.
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Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
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Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
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Prove that a non-cyclic group of order a² has exactly a+3 subgroups. [on hold]

Prove that a non-cyclic group of order a² has exactly a+3 subgroups. Hint: Use Lagrange Theorem
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20 views

Isomorphisim to a cyclic group of prime order [duplicate]

Let G be a nontrivial abelian group whose normal subgroups are only itself and trivial subgroup 1. Prove that G is isomorphic to a cyclic group of prime order.
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34 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
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transitivity of integral extensions

Let $T{\geq}S{\geq}R$ be commutative rings. I'm trying to prove that if $T$ is integral over $S$ and $S$ is integral over $R$ then $T$ is integral over $R$. Let $t$ be in $T$ so there exist ...
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2answers
117 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
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35 views

Classifying $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$

I wish to classify $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$ according to the fundamental theorem of finitely generated abelian groups. We have that it is of ...
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Methods for computing subextensions for a n-th cyclotomic field.

So the problem is 1)find all quadratic and cubic subextensions of $\mathbb{Q}[\zeta^{527}]$ and 2)describe how it's primes split completely in the cubic subextensions. Can you give me some ...
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Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
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9 views

Which of the following Statements are true(CSIR)

Question : Let $R$ and $S$ be non zero commutative rings with unity. Then which of the following statements are true If $S$ is a quotient ring of $R$, then either Char(R) divides Char(S) or ...
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Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ such that $|S\cap T|$ is as larger as possibe and write $D = S \cap ...
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1answer
23 views

Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
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42 views

Irreducibility of a polynomial in two variables

Anyone knows how to verify that the polynomial $$(ax)^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0-y^n$$ is irreducible, where $n\geq 2$, $a,a_i\in\mathbb{Z}$, $a\neq 0$, and $(ax)^n+a_{n-1}x^{n-1}+\cdots ...
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Discriminant of p-adic $\mathbb{Q}_{p}[\phi]$, where $0=f(\phi)=\phi^{p}-\phi-1$

Any suggestions using the minimal polynomial? How about $D_{\mathbb{Q}_{p}[\phi]/\mathbb{Q}_{p}}=(-1)N_{\mathbb{Q}_{p}[\phi],\mathbb{Q}_{p}}(f')$? But foremost I prefer you suggest me a correct ...
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15 views

The Index of an element in a direct product

I'm having a bit of trouble. I know that the index of $a$ in $\mathbb{Z}_m$ is equal to $\frac{m}{|a|}$ Where $|a|$ is the order of $a$ in $m$. But say we have a direct product of $\mathbb{Z}_m ...
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Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$.

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$. Also another question, do p and q have to be distinct for this to hold? On top of ...
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29 views

Finding number of roots of a polynomial in p-adic integers $\mathbb{Z}_{p}$

The problem is to find the number roots of $x^3+25x^2+x-9 $ in $\mathbb{Z}_{p}$ for p=2,3,5,7. I read this equivalent to having a root mod $p^{k}~\forall k\geq 1$. By Newton's lemma I can get whether ...
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Concept for an “Order” of Exponents

Given a set $S$ of abstract mathematical objects, let's say that every element $A\in S$ has order 1, denoted $Ord(A)=1$. I then define that for any 2 elements $A$ and $B$ in $S$, $Ord(A\cdot B) = ...
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Showing quotient rings are isomorphic

Can anyone explain to me how to show two quotient rings are isomorphic? For my particular case. Both quotient rings are based off ideals in the ring $\mathbb Z_3[X]$: $$ \mathbb ...
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1answer
32 views

Fermat's little theorem and congruence classes

I don't understand how to use Fermat's little theorem to find remainders e.g if we are asked to find remainder of $50^{50}$ on division by $13$, what is a and what is $p$ in the formula? Also I ...
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1answer
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Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
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Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...
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Let $G$ be a nonabelian group of order $39$. How many subgroups of order $3$ does it have?

Let $G$ be a nonabelian group of order $39$. How many subgroups of order $3$ does it have? somebody please help me.Thanks for yur kind help.
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Unitary groups deals with matrices how it linked forms ???

But Projective Linear groups and all those things are defined on forms.. Is there any connection between forms and matrices... ??
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$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$

I have done part (i) I Was doing part (ii) and got stuck: Since from above I showed that $H_1 \times H_2 \subseteq K$ now i only need to show that $H_1 \times H_2 \supseteq K$. Let $(g_1,g_2) \in ...
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Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
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1answer
25 views

Find all homomorphism from $S_4$…

I am supposed to find all group homomorphisms from $S_4$ to $\mathbb{Z}_2$, I tried to check if $S_4$ was cyclic then it would be easy but of course it isn't. I am not looking for a solution since ...
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Show that the center of $S_3$ is the identity subgroup.

I know that intuitively this is true, but am not sure how to explicitly show it.
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What does “A mod P generates the residue class field extension” mean?

We have K and finite algebraic extension L. P is a prime ideal in $O_{L}$ over prime $p\in O_{K}$ and $A\in O_{L}$. Then the problem says $\bar{A}:=A ~mod ~P$ generates the residue class field ...
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Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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Is there an equivalent to the DLP with extension fields?

For instance, if I have an extension field of $p^n$, is there a way to recover $p$, other than brute force checking?
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Relation matrix for quotient module

I'm working on a problem: V is an abelian group that can be decomposed by: $V = \frac{Z}{7^3} \oplus \frac{Z}{7^2} \oplus Z$. W is a subgroup of V generated by the image of the element $(7^2,7,7)$ in ...
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45 views

exact sequence proof [on hold]

Let $R$ be a commutative ring and $0\to L\to M\to N\to 0$ be a sequence of $R$ modules. Let $A$ be a multiplicativity closed subset of $R$ so that we can consider the corresponding localisation ...
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1answer
31 views

Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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1answer
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Help with this exercise from Hungerford's book

I'm trying to solve this question from Hungerford's book: The lemma 6.11 says Let $T$ be the subgroup of index $2$, if it generates $A_n$, then $A_n\subset T$, thus by Lagrange theorem: ...
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Prove that G is a group acting on a set X. Where G= {(1),(123),(132),(45),(123)(45),(132)(45)} and X= {1,2,3,4,5}

I understand that the axioms that must be satisfied to prove that this is an "action" is: ex = x for all x an element of X (compatibility with identity). g_1(g_2*x) = (g_1g_2)*x (compatibility with ...
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Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $mr=r'm.$ I want to know if there is a name for this kind of ...
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Prove that $\mathbb Q_p$ and $\mathbb Q_{p'}$ are not isomorphic

Prove that $\mathbb Q_p$ and $\mathbb Q_{p'}$ are not isomorphic as fields for any two distinct primes $p$ and $p'$, where they are denoting the p-adic rationals. I have tried with field isomorphism. ...
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3answers
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the Gaussian integers are isomorphic to $\mathbb{Z}[x]/(x^2+1)$

I am trying to prove that $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$. My initial plan was to use the first isomorphism theorem. I showed that there is a map $\phi: \mathbb{Z}[x] \rightarrow ...
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Every vector space has a basis using minimal spanning set.

We have seen the argument for proving the above statment using Zorn's Lemma by asserting the existence of a maximal linearly independent set which serves a basis. In finite dimensional vector space, a ...
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Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
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1answer
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Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
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Abelian isomorphic group [duplicate]

Prove that if $G$ and $G'$ are isomorphic groups and $G$ is abelian, then $G'$ is abelian, too. Can you please solve this question, I have an exam soon and I have to learn this! I know I asked this ...
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Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.

I've tried proving that $ghg^{-1}\in H$ ($\forall g \in G$), but I don't see how the special property of $H$ guarantees this. Any insight? I've turned away from it to work on other things, and it's ...
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Proving $\dim(\ker( p (T) ) ) = n\cdot d$ where $n$ is a positive integer and $d$ is the degree of the polynomial.

A few more details: $T$ is $T : V -> V$ for some space $V$. Also, the polynomial $p$ is irreducible where $d \ge 1 $. What I've done so far was to restrict the transformation to the invariant ...
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Elements of a Dedekind domain can be chosen to have valuation $1$ with respect to one prime, $0$ everywhere else

I noticed this is true for $\mathbb{Z}$, but I was wondering whether it was true in general. Let $R$ be a Dedekind domain and $P_1, ... , P_s$ maximal ideals. The localized ring $R_{P_i}$ is a ...
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ANT Frohlich Proposition 3, (v). Induced map of dual modules has the same determinant

$R$ is a Dedekind domain, $V$ is an $n$ dimensional vector space over its quotient field $K$, $B(-,-)$ is a $K$-bilinear form on $V$, and $M, N \subseteq V$ are free $R$-modules of rank $n$. I'm ...