# 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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### Determine if the cyclic group $\langle(1234)\rangle$ is normal in $S_4$ [duplicate]

How exactly can I determine if it is normal or not?
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### Proof that field of constants is trivial

Let $d$ be $k$-derivation of $L=k(x_1,x_2,\dots)$ (field of rational functions over field k) defined by $d(x_i)=x_{i+1}$ for $i=1,2,\dots$. Then $L^d=k$. Book says it's easy to prove it but i dont ...
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### Show that A is an algebra

Suppose $X$ is a collection of sets and $Ω$ element of $X$. Also $A$, $B$ are elements of $X$. Then, $A-B=A\cap B^c$ element in $X$. Show that $X$ is an algebra. Please help me, how I can show this ...
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### Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
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### Show that there exists a polynomial $p(X) ∈ F[X]$ satisfying $c^n + c^{−n} = p(c + c^{−1})$.

Let $c$ be a nonzero element of a field $F$ and let $n > 1$ be an integer. Show that there exists a polynomial $p(X) ∈ F[X]$ satisfying $c^n + c^{−n} = p(c + c^{−1})$. I made some particular ...
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### finite dimensional $K$-vector space $V$ with linear endomorphism $T$ is a cyclic module over $K[x]$

Problem: Suppose that $V$ is a finite dimensinal $K$-vector space with a linear endomorphism $T$. Then show that i) Associated $K[x]$-module $V$ is such that $V\cong K[x]/(g)$ for some monic ...
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### Splitting field and polynomial of minimal degree

Let's assume that we have a splitting field $F$ over $Q$ that is a finite extension. Let $p(x)$ be the polynomial in $Q[x]$ that has $F$ as a splitting field and is of minimal degree. Is it correct ...
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### Find the $\ker(f)$ and $\text{Im}(f)$.

Consider the rings $\mathbb{Z}$, $\mathbb{Z}_{4} = \{\bar{0},\bar{1},\bar{2},\bar{3}\}$ and $\mathbb{Z}_{12} = \{[0],[1],[2],...[11]\}$. Define $f: \mathbb{Z} \to \mathbb{Z}_{12}$ by $f(x) = 9x$. ...
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### What do Ideals tell you? [duplicate]

So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is? I believe the definition is: $I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$ ...
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### Polynomial Ring Degree functions

$f(x) = 3x + 2$, $g(x) = 2x^2-3$ in $\mathbb{Z_2}$[x]. What is degree of $f(x)g(x)$? is it simply just the highest degree?
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### Why is the group $Z_6$ under addition isomorphic to the group $Z_7^*$ under multiplication?

Why is the group $Z_6$ under addition isomorphic to the group $Z_7^*$ under multiplication? So I'm trying to answer this question: Q. Which of the set are isomorphic to each other? ...
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### Group Theory Lemma Proof

Suppose that $(G,*)$ and $(H,\circ)$ are groups and $f:G\to H$ is a homomorphism Prove $f(a^n)=f(a)^n$ $\forall a\in G, n\in \Bbb{Z}$ . I can't think of smart ways to manipulate the properties of ...
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### Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
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### Given two arbitrary terms of a geometric series, can this lead to geometric series parameters that cannot be solved for in terms of radicals?

So some higher degree polynomials (and polynomial systems) can still have their roots be solved for in terms of radicals. It comes down to Galois groups and I'm a bit rusty on that subject so I don't ...
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### Manipulations of Euclidean domains

I am trying to answer the following question For (a) I have said that a and ab are in the ring R, by the definition of a ring. Therefore, by the definition of a Euclidean domain a=abq+r. As we are ...
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### Why is the $\mathbb{Z}$ - Hodge conjecture false?

I wish someone well initiated in the area , told me , why the $\mathbb{Z}$ - Hodge conjecture is wrong, which allowed to change its state by tensoring by $\mathbb{ Q }$ to become as it is known ...
Let $R = Z[\sqrt{5}]$ and $f : R → Z$ be defined as $f(x + y\sqrt{5}) := | x^2 − 5y^2|$. Then prove that $f$ is multiplicative : $f(αβ) = f(α)f(β)$, $∀α, β ∈ R$.
Suppose that $R$ is a PID and that $I$ is an an ideal. Then $I$ is maximal iff for any $x$ generating $I$, $x$ is irreducible. This is my try Proof: ⇒: Suppose I is maximal and that I is ...