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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Polynomial systems - conditions for real solution

I was working on the computation of equilibrium points for dynamical systems and arrived in the following system of $n$ polynomials in the variables $(x_1,\ldots,x_n)$ \begin{equation*} ...
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64 views

If every element of a ring is either potent or central, the ring is commutative

Let $R$ be a ring such that every element is potent ($x^k = x$, for some integer $k>1$) or central. Prove that $R$ is commutative. My prove: Let $x,y$ be elements of $R$, suppose one of them ...
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1answer
30 views

Enumerating all bases of vector space of rank n over the finite field $\{0,1\}$

I need to create all bases of a vector space of rank $n$, as efficiently as possible (not going over all $n\text{ x }n$ matrices and deciding if they're a basis). My field is $\{0,1\}$, meaning for ...
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2answers
40 views

The meaning of this definition?

So I am confused with the term "sub-algebra". The book says: if $F$ is a field, and $K$ a subfield, then $F$ is a $K$-algebra. For any set $J\subseteq F$, $K[J]$ is called the $K$-sub-algebra of $F$ ...
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1answer
14 views

parity of function over general field

we know over $\mathbb{R}$, the only function that is both even and odd is the zero function. My question is: what happens if the function is not over $\mathbb{R}$, but over some other field? Field of ...
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2answers
31 views

Is an elementry abelian group a non-degenerate symplectic vector space?

Let $A$ be an elementry abelian group with $|A|=p^{n}$ where $p$ is a prime number and $n$ is even. It is well-known that we can consider $A$ as a vector space of dimension $n$ over the field $F_p ...
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1answer
40 views

$x \rightarrow x^n$ is a group automorphism of a finite abelian group G [on hold]

How do we prove that the map $\phi:G \rightarrow G$ defined by $\phi(x) = x^n$ for some $n \geq 0$ is a group automorphism of $G$ if $\gcd(|G|,n)=1$?
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1answer
41 views

How to prove that a subgroup of a group is normal based on generating sets?

I apologize if this is a duplicate question, but I read online that one method by which to show that a subgroup is normal is by means of generating sets (if both groups have known presentations). In ...
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2answers
29 views

Isomorphism and Cardinality

Is the group $(G, *)$, where $G =\{(x, y) \mid x,y \in \mathbb{R}\}$ isomorphic to $(\mathbb{C}, *)$, where $\mathbb{C}$ is the complex numbers. My initial intuition says no since their are elements ...
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2answers
29 views

Order of a permutation

What does the order of a permutation actually mean? I accept the fact that it is the l.c.m. of the lengths of the cycles in its cycle decomposition, but I don't really have an intuition for what the ...
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1answer
36 views

Write Galois group as semidirect-product

Consider the polynomial $ x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
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45 views

Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
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2answers
20 views

Constructing group homomorphisms.

Let $G$ and $H$ be groups, and suppose I want to construct a group homomorphism $\phi$ between them. From what I know, I just need to send each element $x \in G$ to an element $y \in H$ such that the ...
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0answers
24 views

The Grothendieck group construction

In Lang, the example is stated as follows: Let M be a commutative monoid, written additively. There exists a commutative group K(M) and a monoid homomorphism. $$\gamma: M\rightarrow K(M) $$ having ...
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1answer
17 views

What is the usual definition of a zero divisor?

Let $R$ be a ring. I found there are two distinct definitions: Wikipedia Definition $a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$. Another: ...
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1answer
29 views

Normal extension and group of automorphisms

Let $K \supset F$ a normal extension (finite). Show in details that, for every $\alpha \in K$, the polynomial $$f(x)=\prod_{\sigma \in Aut_F(K)}(x-\sigma(\alpha))\in F[x];$$ My attempt: Since $K$ ...
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40 views

Property of group G with $|G|=2n$ with $n$ elements of order $2$ (Sylow theorem application)

Suppose $G$ is a group such that $|G|=2n$, $G$ has $n$ elements of order $2$ and the rest of the elements form a subgroup $H$. Show that $H \lhd G$ and $n$ is odd. I am pretty lost with this ...
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39 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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1answer
75 views

Question regarding adjoint functors

Suppose $B \rightarrow A$ is a morphism of rings. If $M$ is an $A$-module, one can create $M_B$ by considering it as a $B$-module. This gives a functor $\cdot_B: \mathrm{Mod}_A \rightarrow ...
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1answer
17 views

Non-singular bilinear forms over a vector space.

I have a field $k$ and a finite dimensional $k$-vector space $E$. Let $f$ be a symmetric $k$- bilinear form on $E$. I define $f$ to be non-degenerate if $f(x,y)=0$ $\forall y\in E$ implies $x=0$. I ...
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32 views

Self normalising p sylow

When are p-sylow subgroups self normalising? I know, for example, that if the group has order $ p^2q^2$ then the p-sylow subgroups are self-normalising if there are $q^2$ of them. I just don't know ...
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1answer
36 views

Nonhomogeneous Linear Systems and Vector Space Solutions

Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ...
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1answer
39 views

adjoint of forgetful functor related to localization

Let $A$ be a ring and $S$ a multiplicative subset of $A$ such that $1 \in S$. Let $G$ be the forgetful functor from $Mod_{S^{-1}A} \rightarrow Mod_A$. Taking an $S^{-1}A$-module N and consider it as ...
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22 views

Proof of $g^a \equiv 1 \pmod{m}$ and $g^b \equiv 1 \pmod{m} \quad \implies \quad g^{\gcd(a,b)} \equiv 1 \pmod{m}$

I am trying to understand the following: $$g^a \equiv 1 \pmod m \quad \text{and} \quad g^b \equiv 1 \pmod{m} \quad \implies \quad g^{\gcd(a,b)} \equiv 1 \pmod{m}.$$ I have tried a few approaches, ...
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2answers
37 views

If $|G|<\infty$ and $H\leq G$ is such that $[G: H]=2$ then $|x^G|=|x^H|$ or $|x^H|=\frac{1}{2}|x^G|$ for all $x\in H$?

Let $G$ be a finite group and $H$ a subgroup of $G$ with index $2$, that is, $[G: H]=2$. Recall that $$C_H(x)=H\cap C_G(x), $$ where $C_G(x)=\{g\in G: gx=xg\}$. How can I use the second isomorphism ...
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1answer
20 views

Compute the following permutations [closed]

|(1254)|. I think they are trying to ask of the order of the permutation but I'm not sure how to solve it.
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1answer
44 views

Determining if a set is a group

Let $S=\lbrace x+y\sqrt2 : x,y\in \mathbb R \rbrace$ \ $\lbrace0\rbrace$. Justify whether $S$, together with traditional multiplication, is a group. I've verified that the set is closed under the ...
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28 views

In U(n) how do we find the element such that $x^2=1$ [closed]

For any integer n>2, show that there at least two elements in U(n) such that $x^2=1$
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2answers
115 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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1answer
39 views

Proof that $ \langle x \rangle = \{rx + nx \mid r \in R, n \in \mathbb Z\}.$

I need help in constructing a proof for this: Let M be an R-Module and $x\in M$. Then the submodule generated by x is given by $$\langle x \rangle = \{rx + nx \mid r \in R, n \in \mathbb Z\}.$$
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1answer
61 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
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27 views

find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that ...
2
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1answer
43 views

Goursat's Lemma proof

There is a problem in Lang's book that I don't quite understand how to proceed. It is problem #5, pg 75. I have already shown that the subgroups N and N' can be identified as normal in G, G'. But I ...
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3answers
50 views

Zero divisors and units of $\mathbb Z[X]/ \langle X^3 \rangle$

Problem: Find the zero divisors and the units of the quotient ring $\mathbb Z[X]/\langle X^3 \rangle$. If $a \in \mathbb Z[X]/ \langle X^3 \rangle$ is a zero divisor, then there is $b \neq 0_I$ ...
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1answer
17 views

Are irreducible polynomials and irreducibles (in an integral domain) different?

There's this theorem that you can factor polynomials (in $\mathbb{Z}$[x]) into polynomials of lower degrees r and s in $\mathbb{Q}$[x] iff you can factor that polynomial into polynomials of the same ...
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1answer
28 views

$\varphi(f)$ is invertible iff $f$ is non-degenerate?

Let $E$ be the vectorial space of the bilinear functions $\varphi: \mathbb R^n\times \mathbb R^n\to \mathbb R$. Then, there is a canonical isomorphism between $E$ and the set of the real matrices ...
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27 views

Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
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1answer
48 views

$\mathbb R[X] /<X^2-1>$ and $\mathbb R[X,Y]/<XY>$ are not fields

I have to prove that 1)$\mathbb R[X] /<X^2-1>$, and 2) $\mathbb R[X,Y]/<XY>$ are not fields. So, I must exhibit an element $r$ from say $\mathbb R[X] /<X^2-1>$ that has no ...
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1answer
22 views

Injective endomorphism on a finite field is surjective?

Can you guys give me any hint on how to prove(or disprove): any injective endomorphism on a finite field is also surjective?
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35 views

Isomorphism of a set [duplicate]

We know that $\operatorname {Aut}(G) \over \operatorname {Inn}(G)$ $\cong \operatorname {Out}(G)$. Is it true that $\operatorname {Aut}(G) \cong \operatorname {Inn}(G) \rtimes \operatorname {Out}(G)? ...
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1answer
45 views

automorphism group of groups [closed]

Given a group $G$, I would like to calculate $\operatorname{Aut}(G)$. From definition of $\operatorname{Aut}()$ we know: $\operatorname{Aut}(G)\le \operatorname{Sym}(G) $ If the group is finitely ...
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1answer
60 views

Number theoretic proof that $n\mid\phi(a^n-1)$

While 'playing' with the multiplicative group of integers mod $n$, I noticed that $n\mid \phi(a^n-1)$. The proof is straightforward: $a \in \left ( \mathbb{Z}/(a^n-1)\mathbb{Z} \right ...
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1answer
47 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
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1answer
53 views

Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
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28 views

An Advice Concerning Master's Programme [closed]

Which of these programmes is a better choice, if one wants to pursue a degree in pure mathematics? (In Geometry, Topology and Algebra, in particular, algebraic geometry) 1) ...
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3answers
69 views

Set containing all rings!

Does there exist a set containing all rings ? Possible idea :I think such set is not possible.If S is a set containing all rings i think we can again define a structure on S to make it Ring and that ...
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1answer
36 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
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1answer
11 views

Every finite dimensional representation of an algebra has an irreducible sub representation

Let $V$ be a nonzero finite dimensional representation, i.e we have a homomorphism $\rho\colon A\rightarrow \text{End}_k(V)$, of an algebra $A$. I have to show that there is an irreducible sub ...
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2answers
108 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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1answer
18 views

Statements about ring homomorphisms and division rings

Problem Decide whether the following statements are false or not. 1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, ...