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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Techniques to turn expressions involving integer roots into polynomials by substitution?

Inspired by this question involving an Equation for a Torus How to find a parametrization for a torus? I started wondering if there is some systematic approach to do substitutions to make equations ...
2
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2answers
288 views

prime implies irreducible

In unique factorization ring with unity(I am not considering commutativity and zero divisor in definition of UFD) irreducible implies prime. And it was proved in ring with unity without zero divisor(...
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1answer
26 views

Deciphering the main theorem of the paper ''On Oblath's Problem''

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
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1answer
27 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
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1answer
59 views

Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
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4answers
59 views

Showing that for $f \in K[x]$, we have $f \mid f(x + f(x))$

Let $K$ be a field an $f \in K[x]$. I now want to show that $f(x) \mid f(x + f(x))$ (in $K[x]$). I know that I need to find a polynomial $g \in K[x]$ so that $f(x) g(x) = f(x + f(x))$. So I thought ...
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1answer
36 views

Irreducible elements for a commutative ring that is not ID

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
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2answers
19 views

Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...
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72 views

Have semigroup actions with an additional consistency axiom ever been studied?

For a semigroup $G$ with a left action on itself, the standard axiom for compatibility on some set $X$... $$ \forall x\in X: hg(x)=h(g(x)) $$ ...becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now ...
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1answer
32 views

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$?

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$? It is well known that most (in some suitable sense) polynomials $f \in \mathbb{Q}[x]$ of degree $d$ and coefficients $|...
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2answers
45 views

What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume $x$ is a complex number. $a_k$ are integers. Now consider the corresponding matrix polynomial: $...
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3answers
118 views

Does $K = \mathbb Q/(X^4 - 2)$ contain the imaginary unit $i$?

Let $P(X) = X^4 - 2 \in \mathbb Q[X]$. a) Prove that $P(X)$ is irreducible. b) Prove that the field $K = \mathbb Q/(P(X))$ is an algebraic extension of $\mathbb Q$ and find a generator of it. ...
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1answer
30 views

Showing that $A \otimes_B A \to A$ is a surjective homomorphism.

Let us define a homomorphism $\phi: A \otimes_B A \to A$ by $a \otimes a' \to aa'$ where $A$ is a $B$-algebra, and both $A$ and $B$ are commutative rings. I want to show that this is a surjective ...
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2answers
29 views

If a magma M is both a semigroup and a quasigroup, is it necessarily a group?

If a magma which has an identity element is also a semigroup and a quasigroup, it can be shown that this is indeed a group. I'm looking for a counter example: a magma which is a quasigroup (for every ...
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1answer
52 views

A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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0answers
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Find if an element of $(\mathbb{F}_{2^w})^l$ is invertible

Give an element $a(x) \in (\mathbb{F}_{2^w})^l$ (that I think is a vector space) modulo $l(x)=x^4+1$ with a ring structure because $l(x)$ is reducible, I like to know if this element is invertible and,...
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2answers
69 views

How do I find the ideal $I+J$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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2answers
47 views

Algebra, linear transformation, minimal polynomial [on hold]

Let $T : M_{n×n}(\Bbb F) \to M_{n×n}(\Bbb F)$ the linear transformation defined by $T (A) = AB$, for some matrix $B \in M_{n×n}(\Bbb F)$ fixed. Show that the minimal polynomial of $T$ coincides with ...
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1answer
48 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
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33 views

A problem with infinitely many eigenvalues on a finite dimensional vector space

I want to develop some theory before posing the problem. Kindly stay with me. Consider $ Aut (k[x_1,...,x_n])$ where $k$ is an algebraically closed field, you can take $k=\Bbb C$. $\alpha \in Aut(k[...
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3answers
31 views

Meaning of $Gal(L/L)$ for some field $L$?

In my notes it says $Gal(L/L)=1$ and I am confused on the notation clearly there is only one automorphism of $L$ that map all elements of the base field $L$ to itself namely the identity map. But what ...
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1answer
86 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
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0answers
10 views

Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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1answer
34 views

Problems related to conjugacy classes and their sizes.

Is $Z( G_1 \oplus G_2 \oplus G_3 \oplus\cdots\oplus G_n) = Z(G_1) \oplus Z (G_2) \oplus Z(G_3) \oplus \cdots \oplus Z$ $(G_n)$ true, where $ G_1, G_2, G_3 \cdots G_n$ are finite groups?$Z$,here refers ...
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0answers
30 views

Congruence numbers

Having read about Stirling numbers of the second kind I am curious. The article says it shows the number of equivalence relations on a set $n$ with $k$ equivalence classes which makes sense to me from ...
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galois group of a palindromic polynomial is not $S_n$?

Let $f(x) = a_nx^n+....+a_0 \in \mathbb{Q}[x]$ be a palindromic polynomial; that is, the coefficients of $f$ satisfy $a_n = a_0, a_{n-1} = a_1$, and more generally $a_{n-i} = a_i$ for all $0\leq i\leq ...
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1answer
21 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
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1answer
22 views

compact metric space definition by closed covers

My book says the following: A metric space is compact iff: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\lambda}$ is open. Then, it says that if $A_\...
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0answers
40 views

Solve the nth zero of a function. [on hold]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
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2answers
71 views

Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
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22 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
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2answers
214 views

Normal subgroup in center of the group

Let $G$ be a group of order $3825$. Prove that if $H$ is a normal subgroup of order $17$ in $G$ then $H\leq Z(G)$. In the link below, the solution basically says that the index of $C_{G}(H)$ must ...
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5answers
57 views

Is $F[x,y]$ a Euclidean Domain?

I was wondering if this is just common knowledge. So far for a field $F$ and transcendental $x$ and $y$, I know one can define the degree by $1) \deg c =0$, for any $c \in F-\{0\}$ $2) \deg x^{n_1}...
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2answers
113 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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1answer
52 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
1
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1answer
40 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
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Primitive solvable group

Let $G$ be a finite solvable group. Suppose that $G=HN$ for all minimal normal subgroups $N$ of $G$. To show that $H = G$ or $G$ is primitive If $N$ is a minimal subgroup of $G$ then $N$ is an ...
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3answers
22 views

Misunderstanding the definition of a cycle (cyclic permutation)

Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$. Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\...
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1answer
52 views

Show that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$

Let $R$ be a commutative ring with unit and let $M$ be a $R$-module. It holds that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$, right? How could we prove this? Do we have to define ...
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4answers
167 views

Element of infinite order for a given group presentation

Let $G=\langle a,b,c,d \mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle$ be our presentation. The claim is that the commutator $[a,b]$ has inifinite order in $G$. I think this might be related to small ...
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1answer
120 views

Minimal polynomial for $x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$

I found numerically that the minimal polynomial for: $$x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$$ has the following form: $$(3p^2-1)(p^2-1)\color{blue}...
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504 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
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2answers
145 views

An example of a ring without identity that does not contain any maximal ideal. [duplicate]

I'm trying to find an example of a ring without identity that does not contain any maximal ideal. Help me some hints.
3
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1answer
397 views

if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.

I'm having trouble with this homework problem (from Algebra by Hungerford). If $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. ...
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1answer
136 views

$R$ be a ring without identity. If $R$ has a maximal left ideal, then the Jacobson radical is still the intersection of all the maximal left ideal?

We know that the definition of the Jacobson radical $J(R)$ (a) in a ring $R$ with identity is: $$J(R)=\cap \mbox{ maximal left ideals}.$$ (b) in a ring $R$ without identity is: $$J(R)=\{a\in R\mid ...