Tagged Questions

6
votes
0answers
57 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
2
votes
2answers
91 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
6
votes
1answer
46 views

Given $G$, when can we find a division ring $R$ with $R^*=G$?

This is motivated by a characterization of finite cyclic groups, in which one proves Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic. The proof is ...
2
votes
1answer
30 views

Characteristic of commutative semisimple rings?

In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
1
vote
1answer
33 views

Field, Euclidean division question.

Let $K$ be a field and $f \in K[x]$. Show that if there is some $a \in K$ such that $f(a)=0$, then $x-a$ divides $f$. My friend told me to use Euclidean division by $x-a$. Also show that a ...
6
votes
3answers
79 views

Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
3
votes
0answers
58 views

Hilbert’s zeros theorem, an application. (The algebraic variation)

Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
1
vote
1answer
57 views

Field extension of $\mathbb Q$ of degree 2

Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$. I know that $K$ is generated by say ...
3
votes
3answers
136 views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
1
vote
1answer
33 views

Finitely generated integral domain and finitely generated $k$-algebra.

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
6
votes
1answer
83 views

Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
3
votes
2answers
141 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
1
vote
1answer
48 views

Build Onto mapping of a Field to a field with an Integral Domain

The question is as follows, Let D be an integral domain. Let Q be its field of fractions and $\phi$ : D --> Q be the canonical map of D into Q. Prove that, if D is a field, then $\phi$ is ...
4
votes
2answers
92 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
1
vote
1answer
74 views

Ring/Field isomorphism before knowing it's a ring/field

If I know $F$ is a ring/field, and I have $G$ (unkown), but I can find a bijective map: $$\Phi:F\to G$$ such that $\forall a,b\in F.\;\Phi(a+b) =\Phi(a)+\Phi(b).\;\Phi(ab) = \Phi(a)\Phi(b)$ Does that ...
5
votes
6answers
294 views

Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
6
votes
3answers
221 views

How to show that a finite comutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
1
vote
2answers
61 views

Subfields of Rings

I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about. Can a ring that is not a field, have a subring that satisfies the ...
0
votes
2answers
71 views

Splitting of polynomial into linear factors

I am trying to show that the polynomial $x^3 - 3$ splits into linear factors over $ \mathbb{Z}_7[x]/ \langle x^3 - 3 \rangle $, but am having trouble doing so, as I'm not very familiar with rings. ...
1
vote
1answer
128 views

Irreducible polynomial over field

I have a problem to prove that $x^5+4x^4+10x^2-6x+2$ is Irreducible polynomial over the field $\mathbb{Q}(7^{(1/7)})$ I try to use eisenstein criterion for the polynomial above ...
3
votes
2answers
85 views

Lost on algebra notation

I'm in a basic number theory course and, never having taken college algebra, I'm lost on some of the notation. I'm wondering what some of these notations mean: 1: $\mathbb{Q}(i)$ ... I know that what ...
2
votes
1answer
173 views

A finite field extension that is not simple

Let $F=\mathbb{F}_p(X,Y)$ be the field of rational functions in variables $X,Y$ over the finite field of $p$ elements. Let $K=\mathbb{F}_p(X^p,Y^p)$ be a subfield. Note that for any $f\in F$, $f^p\in ...
1
vote
1answer
74 views

Conjecture regarding primal representaion of vectors

I have a conjecture which I cannot prove or disprove. Denote the $i$'th digit of $x$ in binary expansion by $d_i(x)$, where for $i=1$ the MSB is taken. Example: $d_3(110.1)=0$ and $d_4(110.1)=1$ (so ...
7
votes
1answer
166 views

Two questions on Nagata's counterexample to the Hilbert's fourteenth problem.

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
1
vote
1answer
44 views

Fields arising as endomorphism rings

Do you know a field $K$ other than $F_p$ which is the endomorphism ring of an abelian group $G$? I doubt that there is one because as $G$ gets bigger, $End(G)$ seems to be more and more ...
0
votes
1answer
69 views

Variation of the universal property for the field of fractions

Consider the universal property for the fraction field of an integer domain: Let $R$ be a integral domain, $F(R)$, its fraction field, $K$ some field and $f:R\rightarrow K$ a injective ring ...
1
vote
3answers
111 views

field of prime characteristic

Say $F$ is a field of characteristic $p$ and let $f(x) = x^p - a \in F[x]$. Show that $f$ is irreducible over $F$ or $f$ splits in $F$. Well, my solution would be since $Char F = p$, then $(x - ...
1
vote
1answer
78 views

Abstract Algebra elementary question

Let $F = \mathbb{Q}(\pi^3)$. How can we find a basis for $F(\pi)$ over $F$ ?
0
votes
2answers
67 views

Relation between torsion subgroup of multiplicative group of field and solvability of polynomials

In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
0
votes
1answer
78 views

Subring of a Field with characteristic P, a prime number.

Suppose $F$ is a field with characteristic $p$, a prime number. Prove that $F$ contains a subring identical to $Z_p$. Are identical subrings the same? There is no mention of this in the text. So do ...
4
votes
1answer
58 views

Levels of Rings and Fields, -1 as a sum of squares

Definition: Let $R$ be a commutative ring. The level of $R$, denoted $s(R)$, is the least positive integer $s$ such that $-1$ can be written as the sum of $s$ many squares in $R$. Set $s(R)=\infty$ if ...
1
vote
1answer
56 views

Extension of some properties of $\mathbb{R}$ to other fields and subrings.

We know that the only non-zero ring homomorphism from $\mathbb{R}$ to $\mathbb{R}$ is identity. From this some questions came in to my mind as follow: Question $1$: Can we characterize all fields ...
3
votes
1answer
74 views

Equivalence of Valuations - Trouble Understanding Proof

I want to complete the proof of the following theorem. Here is what I have got so far: Theorem Every non-euclidean valuation $v$ on a number field $K$ is equivalent to $v_{\mathfrak p}$ for some ...
1
vote
1answer
51 views

A question regarding solutions of polynomials in a field

Let $F$ be a field and $\langle a_1,...,a_n \rangle \subset F$. Then given a non-zero polynomial $f \in F[X_1,...,X_n]$ is it true that if $f(a_1,...,a_n)=0$ then $(X_i - a_i)$ divides $f$ for some ...
3
votes
2answers
108 views

Understanding of extension fields with Kronecker's thorem

In the book Contemporary Abstract Algebra by Gallian it defines an extension field as follows: A field $E$ is an extension field of a field $F$ if $F\subseteq E$ and the operations of $F$ are ...
1
vote
1answer
73 views

Let $L/K$ be a field extension. Must a $K$-homomorphism $\theta: L\rightarrow L$ be an isomorphism?

I am thinking about $\mathbb{C}/\mathbb{Q}$. But other than the identity map and taking complex conjugates, I cannot think of anything. Any ideas? Thanks. Edit I just managed to show that if $L/K$ is ...
5
votes
0answers
217 views

System of polynomial equations over rational field

Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations: ...
0
votes
2answers
57 views

Is $t^{2}$ a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$?

I wish to find out if $t^{2}$ is a prime element of $\mathbb{F}_{2}(t^{2},s^{2})$ so I can justify the use of Eisenstein on the polynomial $x^{2}-t^{2}\in\mathbb{F}_{2}(t^{2},s^{2})[x]$ I believe ...
0
votes
2answers
44 views

Irreducibility of $x^{3}-t\in\mathbb{C}(t)[x]$

Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$ Is it true that $p$ is irreducible over $F$ ? My thoughts: I think that since it is not true that $t^{2}\mid t$ (I don't know how to ...
4
votes
3answers
165 views

fields are characterized by the property of having exactly 2 ideals [duplicate]

Possible Duplicate: A ring is a field iff the only ideals are $(0)$ and $(1)$ Michael Artin's Algebra in the introduction of maximal fields, there was a sentence stated that fields are ...
1
vote
2answers
118 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
2
votes
1answer
136 views

What is wrong with this proof of Wedderburn's little theorem?

Wedderburn's little theorem $\quad$ every finite domain $A$ is a field. Proof $\quad$ Let $x$ be a nonzero element of $A$. Because $A$ is finite, there exist positive integers $n$, $k$ such that ...
2
votes
0answers
161 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
-3
votes
1answer
127 views

abstract algebra [closed]

How many of the following statements are false? a) Subring of a ring is a ring. b) Subring of commutative ring is a commutative ring. c) Subring of a integral domain is an integral domain. d) Subring ...
3
votes
3answers
152 views

Algebraic Elements and Fields of Quotients

The algebraic elements of $\mathbb{R}$ are those elements which are roots of nonzero polynomials with coefficients in $\mathbb{Q}$. In fact, by multiplying through by denominators, we can even take ...
4
votes
2answers
140 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
3
votes
1answer
116 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
32
votes
4answers
1k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
0
votes
2answers
88 views

Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.
7
votes
3answers
179 views

When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?

I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$. I'm trying to figure out if they're isomorphic (as rings I ...

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