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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.

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46 votes
8 answers

Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....'s user avatar
166 votes
3 answers

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using elementary ...
user8465's user avatar
  • 1,763
85 votes
7 answers

Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{...
user avatar
55 votes
6 answers

Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem: If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
user avatar
152 votes
41 answers

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like this: ...
Sev's user avatar
  • 2,123
132 votes
9 answers

Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Sigur's user avatar
  • 6,466
92 votes
4 answers

If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is cyclic, ...
Altar Ego's user avatar
  • 5,352
103 votes
8 answers

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
MathTeacher's user avatar
  • 1,549
19 votes
5 answers

$\gcd(a,b,c)\!=\!1\Rightarrow \gcd(az+b,c)\!=\! 1$ for some $z$ [Coprime Dirichlet Theorem]

I can't crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$
Vees's user avatar
  • 511
114 votes
7 answers

Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
QETU's user avatar
  • 1,149
61 votes
5 answers

Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
rupa's user avatar
  • 693
34 votes
9 answers

Units and Nilpotents

If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit. I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) = 1 ...
Shannon's user avatar
  • 1,341
72 votes
3 answers

Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots +...
user avatar
158 votes
1 answer

Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$. $(...
Makoto Kato's user avatar
  • 42.8k
54 votes
3 answers

Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
palio's user avatar
  • 11.1k
49 votes
15 answers

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem ...
Siyanda's user avatar
  • 2,559
15 votes
2 answers

Why polynomial division algorithm works for $x-a$ or any monic polynomial?

In Theorem 5.2.3 in these notes, it is said that Since $x − a$ has leading coefficient $1$, which is a unit, we may use the Division Algorithm... Why is this true? I thought that the Division ...
Dominic's user avatar
  • 153
21 votes
3 answers

Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$

I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q}[x]$. Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having ...
Andy's user avatar
  • 243
76 votes
7 answers

Is an automorphism of the field of real numbers the identity map?

Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map? If yes, how can we prove it? Remark An automorphism of $\mathbb{R}$ may not be continuous.
Makoto Kato's user avatar
  • 42.8k
49 votes
2 answers

Group of even order contains an element of order 2

I am working on the following problem from group theory: If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd. That is, for some integer $k$, there are $2k+...
Jacobsen's user avatar
  • 507
40 votes
5 answers

Why are polynomials defined to be "formal" (vs. functions)?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
user avatar
22 votes
7 answers

$I+J=1 \Rightarrow I^n+J^n = 1$ for ideals (or elements in GCD domain) [Freshman's Dream Binomial Theorem]

Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\...
user avatar
56 votes
8 answers

$\langle 2,x \rangle$ is a non-principal ideal in $\mathbb Z [x];\, $ $D[x]$ PID $\iff D$ field, for a domain $D$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is $\...
Person's user avatar
  • 703
26 votes
2 answers

When $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a \in F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
Questions-Math's user avatar
68 votes
6 answers

Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then ...
Melesia's user avatar
  • 681
83 votes
3 answers

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
spin's user avatar
  • 12k
58 votes
3 answers

$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing I know which connects a group ...
user avatar
55 votes
8 answers

Examples and further results about the order of the product of two elements in a group

Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$? Wikipedia says "not much": There is no general formula relating the order of a ...
Jacopo Notarstefano's user avatar
48 votes
6 answers

Group of order 15 is abelian

How do I prove that a group of order 15 is abelian? Is there any general strategy to prove that a group of particular order (composite order) is abelian?
Mohan's user avatar
  • 15k
6 votes
4 answers

Efficiently prove $2$ generates $(\mathbb{Z}/19\mathbb{Z})^*$ [Order Testing]

So I'm first asked to compute, mod 19, the powers of 2, $$2^{2},2^{3},2^{6},2^{9}$$ which I compute as $$4,8,7,18$$ respectively. I'm then asked to prove that 2 generates $(\mathbb{Z}/19\...
Addem's user avatar
  • 5,696
59 votes
2 answers

$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?

Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]: Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in F$. Then $x^...
user79709's user avatar
  • 593
50 votes
6 answers

Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.

Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. I need this theorem to prove something.
BBred's user avatar
  • 591
32 votes
5 answers

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\...
DBr's user avatar
  • 4,820
1 vote
3 answers

Isomorphisms $\mathbb{Z}/n \mathbb{Z}\cong \mathbb{Z}_n$ and $K[x]/(f)\cong K[x]\bmod f\ $ [transport quotient ring structure to remainders]

Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}_n$? Is there any good homomorphism $\phi$ I could use that graphs $\mathbb{Z}/n\mathbb{Z}$ to $\...
Akaichan's user avatar
  • 3,454
8 votes
2 answers

$a^{\phi (n) +1} \equiv a \pmod{\! n}; $ Carmichael generalization of Fermat & Euler theorems.

I want to know a proof of an alternative form of Fermat-Euler's theorem $$a^{\phi (n) +1} \equiv a \pmod n$$ I searched some number theory books and a cryptography book and internet, but there were ...
quicksilver's user avatar
124 votes
13 answers

Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however,...
71 votes
2 answers

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
user avatar
95 votes
6 answers

Why can't the Polynomial Ring be a Field?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This ...
IAE's user avatar
  • 1,347
93 votes
6 answers

Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$?

Is $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbb{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbb{Q}\}$$ $$\mathbb{Q}(\sqrt{2}+\sqrt{3}) = ...
Tashi's user avatar
  • 1,633
45 votes
5 answers

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=...
Mohan's user avatar
  • 15k
20 votes
5 answers

Are associates unit multiples in a commutative ring with $1$?

Recall the following relevant definitions. We say that $b$ is divisible by $a$ in $R$, or $a\mid b$ in $R$, if $b = r a$ for some $r\in R$. $a$ and $b$ are associates in $R$ if $a\mid b$ and $b\mid ...
ShinyaSakai's user avatar
  • 7,906
90 votes
2 answers

Is Lagrange's theorem the most basic result in finite group theory?

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ...
lhf's user avatar
  • 217k
69 votes
2 answers

Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able ...
Danielle Intal's user avatar
53 votes
5 answers

A commutative ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
Jackson Hart's user avatar
  • 1,610
51 votes
3 answers

For what $n$ is $U_n$ cyclic?

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is cyclic? $$U_n=\{a \in\mathbb Z_n \mid \gcd(a,n)=1 \}$$ I searched the internet but did not get a clear idea.
Sankha's user avatar
  • 1,405
41 votes
6 answers

Do odd imaginary numbers exist? [parity for Gaussian integers]

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
InterestedGuest's user avatar
55 votes
5 answers

How to prove that the sum and product of two algebraic numbers is algebraic? [duplicate]

Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \...
spin's user avatar
  • 12k
39 votes
4 answers

Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?

In this post we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? I know these as $\mathbb{Q}$-Vector spaces, ...
user avatar
48 votes
1 answer

Why isn't an infinite direct product of copies of $\Bbb Z$ a free module?

Why isn't an infinite direct product of copies of $\Bbb Z$ a free module? Actually I was asked to show that it's not projective, but as $\Bbb{Z}$ is a PID, so it suffices to show it's not free. I ...
lee's user avatar
  • 2,820
57 votes
2 answers

Why is the ring of matrices over a field simple?

Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...
awllower's user avatar
  • 16.6k

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