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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

Abstract algebra is the study of algebraic objects, i.e. sets endowed with one or more operations on the elements of those sets. In particular, the study of abstract algebra considers the algebraic structures and properties of which such operations induce. It can be considered as the generalization of the study of the algebraic structure of the integers and real numbers (arithmetic), or the study of matrices and vector spaces (linear algebra).

Some algebraic objects are monoids, groups, rings, fields, vector spaces, modules, algebras, and categories, among many other less prominent objects.

Examples

  1. The set of non-negative integers $\mathbb{N} = \{0,1,2,3,\dotsc\}$ is a monoid under the operation $+$.

  2. The integers $\mathbb{Z} = \{\dotsc,-1,0,1,\dotsc\}$ under the binary operation of $+$ form a group.

  3. Furthermore, $\mathbb{Z}$ has the structure of a ring when you consider it as being equipped with both addition and multiplication.

  4. The real numbers $\mathbb{R}$ with their usual addition and multiplication form a field.

  5. The set of $n\times n$ matrices with entries in $\mathbb{R}$ with matrix addition and multiplication form a ring.

  6. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

85022 questions
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$I[X]$ is a prime ideal in $R[X]$ iff I is a prime ideal in $R$

Let $R$ be a commutative ring (not necessarily unital). Let I be an ideal of R. Show that $I[X]$ is a prime ideal in $R[X]$ iff I is a prime ideal in $R$. I have attempted to use the facts: I is prime iff $R/I$ is an ID. I is maximal iff $R/I$ is…
user71346
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Construct a field having 49 elements

I am working on this problem from Herstein's book, Abstract Algebra, 3rd edition. One question asks to construct a field of 49 elements. It gives a hint to use the ring of Gaussian integers and a maximal ideal. This is what I have so far. All rings…
ldiaz
  • 104
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Homogeneous elements and graded algebra - any misconception?

Elements of any factor $A_n$ of the decomposition are known as homogeneous elements of degree $n$. An ideal or other subset $\mathfrak{a} ⊂ A$ is homogeneous if every element $a ∈ \mathfrak{a}$ is the sum of homogeneous elements that belong…
Maps from subways
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Prove the irreducibility of a polynomial

Let $p(x) = x^2 + ax + b$ a polynomial with $a,b\in\Bbb Z$ odd integers. Prove that $p$ is irreducible at $\Bbb Z[X]$, and at $\Bbb Q[X]$. I know the Eisenstein's criterion but I'm not sure about how to apply it.
Oriol Sors
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$\mathbb{Z}_n$ is not a subring of $\mathbb{Z}$

I come across an example stating that ' $\mathbb{Z}_n$ is not a subring of $\mathbb{Z},n \geq2, n \in \mathbb{Z}$' in the book ' Dummit and Foote , abstract algebra'. Can anyone explain to me why the statement is true?
Idonknow
  • 15,643
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Solve an equation in a cyclic group

Let $G$ be a finite cyclic group of order $n$. If $d$ is a positive divisor of $n$, prove that equation $x^d = e$ has exactly $d$ distinct solutions in $G$ This is practice question assigned to me and I have no clue how to approach this.
Kj Tada
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Question on computing direct limits

Lately, I've been working on direct limits. In particular, given $$\mathbb{Z}^n \xrightarrow{M} \mathbb{Z}^n \xrightarrow{M} \mathbb{Z}^n \xrightarrow{M} \cdots$$ where $M$ is an $n \times n$ matrix over $\mathbb{Z}$, then (1) if the eigenvalues of…
Eric
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Question about irreducible polynomials over finite fields

I have the polynomial $f(T)=T^2+T+1$; then, for which primes $p$ does $f(T)$ have roots in $\Bbb F_p$? I tried this way: since the three roots of $f(T)$ are generated from the cubic root of $1$, we need it to be contained in the field $\Bbb F_p$;…
Dr. Scotti
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Is $\frac{\mathrm d}{\mathrm dx}$ an operation?

What exactly is an operation? I understand that multiplication and addition are operations, but what about the derivative ($\frac{\mathrm d}{\mathrm dx}$). Can an operation be a relation of expressions, or just of numbers?
Frasch
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Show the nilradical of $F_p[x] \otimes_{F_p[x^p]}F_p[x]$ is principal

Here are my thoughts: Suppose $F_p[x]$ is flat over $F_p[x^p]$. Then applying $- \otimes_{F_p[x^p]}F_p[x]$ to the natural injection $F_p[x^p] \to F_p[x]$ shows that the map $F_p[x] \to F_p[x] \otimes_{F_p[x^p]}F_p[x]$ given by $f \to f \otimes 1$ is…
John
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Irreducibility of the polynomial $f (x) = x^{2p} − x ^p + t$

If $k$ is a field of characteristic $p > 0$ and $f (x) = x^{2p} − x ^p + t \in k(t)[x]$, how can we show that $f (x)$ is an irreducible polynomial in $k(t)[x]$ and that $f (x)$ is inseparable? If $f(x)$ is irreducible then $D_xf(x) = 0 \implies( f,…
jim
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Maximal ideals in polynomial rings with real and complex coefficients

I was asked in homework to think about maximal ideals in polynomial rings $\mathbb{R}[x]$ and $\mathbb{C}[x]$. I have realized that: $\forall c\in\mathbb{R},\;I_c : = \{p(x)\in\mathbb{R}[x]\;|\;p(c) = 0\}$ is an ideal (similar for $\mathbb{C}[x]$),…
mez
  • 10,497
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Suppose that $m,n \in \mathbb{Z}$ and $m$ divides $n$. Show that $\frac{\mathbb{Z}_n}{\mathbb{Z}_m} \cong \mathbb{Z}_\frac{n}{m}$

Suppose that $m,n \in \mathbb{Z}$ and $m$ divides $n$. Show that $$\frac{\mathbb{Z}_n}{\mathbb{Z}_m} \cong \mathbb{Z}_\frac{n}{m}$$ I try to use the third isomorphism theorem to show but I don know how to apply it here. Anyone can guide me ?
Idonknow
  • 15,643
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"Lifting" an algebraic structure on a codomain to the set of functions into it.

Let $ S$ be any set and $G$ be a group. The set of functions from $S$ to $G$ is clearly also a group. The identity element is the constant function whose value is the identity of $G$, and the (group) inverse of any function $g$ is the function that…
dxuhuang
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Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$. Prove that $x^2=e$ and $y^8=e$ for all $y\in G$.

To put this in context, this is my first week abstract algebra. Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$. Prove that $x^2=e$ and $y^8=e$ for all $y\in G$. A hint would be appreciated.
Kasper
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