# 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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### Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.
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### Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field

I want to show that $\mathbb Z[x] /<5,x^3+x+1>$ I am currently self-studying and it is a problem of previous graduate school entrance exam. I studied abstract algebra through Fraleigh's book, ...
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### Problem in solving a question related to symmetric group. [closed]

The question is : For $n \geq 4$ prove that $S_n$ the symmetric group is generated by $n-1$ elements of order $2$. How can I solve it?Please help me.Thank you in advance.
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### How does Gauss's lemma follow from Nagata's lemma?

In section 4 of Samuel's Unique Factorization it's said Gauss' lemma is an easy consequence of Nagata's lemma. How does this work, i.e., how to deduce Gauss' lemma from Nagata's lemma? I'm asking ...
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### Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $\mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $x^5 - x + 1$. If we let $\zeta$ be a root of this ...
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### field trace function is linear

Assume $E / F$ is a finite extension, the trace function is defined as $$\operatorname{Tr}(a_1)=[E:F(a_1)](a_1+a_2+\ldots+a_n)$$ (where $a_j$ are all of the roots of $\min(a_1,F)$). Then, what I want ...
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### a simple problem about commutative algebra [duplicate]

I think this problem seem easy, but I have no idea to approach this. Let $R$ is commutative Ring, a non-zero $f\in R[X]$ is zerodivisor if and only if there exists no-zero $c\in R$ such that $c.f=0$...
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### Why is a projective module called “projective”? [duplicate]

Is it related to projections as in $P P = P$?
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### Example of a non-Kummer totally tamely ramified Galois extension

Let $A$ be a DVR with fraction field $K$, and let $L$ be a totally tamely ramified finite Galois extension of $K$ of degree $e$ - ie, the integral closure $B$ of $A$ in $L$ is a DVR with ramification ...
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### How to make check matrix H when you have generator matrix (algorithm)

It's all built on top of python numpy lib. So we have a class finite field and get access to elements of field like Finite_field[index_of_element]. Elements of field are numpy matrices(ndarray). For a ...
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### Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$?

Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$? I have a feeling that it's a rather easy question, but since my understanding of field extensions is only superficial I really can't handle this ...
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### Properties of Short Exact Sequences

Some of the work I have been doing lately is heavily dependent on chasing commutative diagrams so I have been brushing up on short exact sequences since I was not familiar with them. For the most part ...
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### $\mathbb{T}\text{-Alg(Set)}$ is complete and cocomplete.

Let $\mathbb{T}$ be a finitary algebraic theory and $\mathbb{T}\text{-Alg(Set)}$ be the category of finite-product-preserving functors $\mathbb{T} \rightarrow \text{Set}$. It is written in my ...
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### Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
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### Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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### Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
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### why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]

Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
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### Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
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### Isomorphism between a quotient of a polynomial ring and a polynomial ring [on hold]

I'm going to show that $\mathbb{K}[x,y,z]/(y^2-xz)$ is not isomorphic to any polynomial ring. I'll be grateful of someone brings a hint to show this result. Thank you.
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### Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.

I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!
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### To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$\bigcap_{N \in \mathcal{N}} N \ = \ \{ e\}$$ I know how free ...
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### Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $S^1 \times S^1 \times S^1$? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
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### About the definition of polynomials on vector spaces?

In the book Linear Systems Theory and Introductory Algebraic Geometry (R. Hermann) the author defines A polinomial on V (a $\mathbb K$-vector space) is an element of the smallest ...
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### Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...