1
vote
0answers
10 views

Projective special linear groups

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic, is co-hopfian? I've searched everywhere but found nothing. Definition A group $G$ is called co-hopfian if $G$ has no ...
0
votes
1answer
40 views

Additive inverse in a field is unique

there will be an $a$ let's prove $-a$ is unique. let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$ let's multiply them by the multiplicative inverse of $a$ I try to use the ...
2
votes
1answer
35 views

A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
1
vote
1answer
77 views

Describing $\mathbb{C}[x,y] / (y^2-x^3)$

First, I'd like to show $\mathbb{C}[x,y] / (y^2-x^3)$ is an integral domain. Then I need to find out whether or not it is a PID. For the first part, I want to show $y^2-x^3 \: | \: fg \implies ...
6
votes
5answers
250 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
1
vote
2answers
33 views

$K \le B\le F$, If $F$ Galois over $K$ $\Rightarrow$ $F$ a Galois over $B$

Let $F$ a Galois over $K$, and let $B$ be a subfield of $F$ such that : $K \le B\le F$ $\Rightarrow$ $F$ a Galois over $B$ PROOF: $F$ is a splitting field of $f \in K[x]$ separable over $K$. ...
5
votes
1answer
49 views

$F/K$ algebraic and every nonconstant polynomial in $K[X]$ has a root in $F$ implies $F$ is algebraically closed.

Let $F/K$ be an algebraic extension of fields in characteristic zero. If $F/K$ is normal, and every nonconstant polynomial $f \in K[X]$ has a root in $F$, then $F$ is algebraically closed. This is ...
5
votes
3answers
174 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
1
vote
0answers
40 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
0
votes
3answers
55 views

For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?

sorry if this question seems obvious, For a field F and $a,b\notin F$, does $[F(a,b):F(a)]=[F(b):F]$? If so, how do you prove it or is there a counter example?
1
vote
2answers
95 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
3
votes
3answers
96 views

Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
3
votes
1answer
69 views

Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
1
vote
1answer
69 views

Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...
1
vote
1answer
38 views

Degree of a single extension of a field. [closed]

Let $K$ be a field. Let $L:=K(b)$ where $b^m\in K$ but $b^i\not\in K$ for all $0<i<m$. Then is $[L:K]=m$?
1
vote
1answer
64 views

Characterize elements in $\mathbb Q(\sqrt2,\sqrt[3]5)$

I'm studying algebra by Herstein's Topics in Algebra, 2ed, and stuck at Ex.5.1.6(b): In $\mathbb Q(\sqrt2,\sqrt[3]5)$ characterize all the elements $w$ such that $\mathbb Q(w)\ne \mathbb ...
4
votes
2answers
40 views

Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective?

Why is $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism given $\phi : F[x] \rightarrow F(a)$ satisfy $\ker \phi = \{0\}$ ? I've been trying to figure out why $\bar \phi$ is an isomorphism, and ...
1
vote
1answer
51 views

How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
2
votes
1answer
87 views

Radical Solvable quintic polynomial.

I am trying to solve the following exercise: Let $k$ be a field of characteristic 0, and let $f(x)\in k[x]$ be a polynomial of degree 5 with splitting field $E/k$. Prove that $f(x)$ is ...
2
votes
1answer
44 views

$(u+1)(u+2)\cdots(u+p)=u^p-u$

Let $E$ be a field with characteristic $p>0$. How should I prove that $$(u+1)(u+2)\cdots(u+p)=u^p-u$$ I can verify this equality for small $p=2,3,5$. Is there a way to prove this result in general? ...
3
votes
3answers
222 views

What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...
23
votes
1answer
239 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
0
votes
1answer
35 views

A question about separable extension

We say $K$ is separably generated over $k$ if there exists a transcendence basis $\{x_i,i\in I\}$ of $K/k$ such that $K/k(x_i,i\in I)$ is a separable algebraic extension. We say $K$ is separable over ...
5
votes
1answer
52 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
3
votes
1answer
45 views

what is this property called?(Field theory)

In what references the following property $P$ of a field $F$ is investigated? The property $P$: For all $n\in \mathbb{N}$ if $\sum_{i=1}^{n} f_{i}^{2}=0,\;\;f_{i}\in F$ then $f_{i}=0, \forall i\in ...
0
votes
1answer
47 views

Possible Fields?

Is there an algebraically closed field which is a 1-dimensional vector space (as opposed to complex numbers which are 2-D)? Also is there a complete $\aleph_0$ field?
3
votes
0answers
26 views

Why is the subfield (of a field) generated by an algebraic element equal to the subring generated by the same element?

I am trying to prove that, for a field extension $\mathbf{K}/k$ and $a$ an algebraic element over $k$, $$k(a)=k[a],$$ where $k(a)$ is the subfield of $\mathbf{K}$ generated by $a$ and $k[a]$ is the ...
1
vote
1answer
39 views

A field extension of degree 8

I would really appreciate it if you give me a hint on the following question: If $K \subset F$ is a field extension of degree 8, then we must have $F=K(a,b,c)$ for some a, b and c in F.
5
votes
2answers
81 views

Dimension of $\Bbb Q(e)$ over $\Bbb Q$?

The dimension of $\Bbb Q(\sqrt{2})$ over $\Bbb Q$ is finite since $\sqrt2$ is algebraic over $\Bbb Q$. But what about any transcendental number (say $e$)? Which is the smallest field containing $\Bbb ...
2
votes
1answer
24 views

Field Isomorphisms between a field and something that contains it

Are there any k-isomorphism of fields between M and L such that K $\subseteq$ M $\subset$ L? Examples would be appreciated. Thanks
23
votes
6answers
1k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
2
votes
2answers
35 views

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
6
votes
2answers
382 views

Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

$F$ is a field of order $32$. $F$ and {$0,1$} are trivial subfields of $F$. But how can we show that these are the only subfields of $F$? Can someone give me a direction to this question?
0
votes
1answer
37 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
9
votes
2answers
313 views

Is there a 'conjugation' on every algebraically closed field?

Let $K$ be an algebraically closed field. Then the polynomial $x^2+1\in K[x]$ has two distinct roots (when $K$ doesn't have characteristic 2). Let's suggestively call them $i$ and $-i$. Does there ...
-1
votes
2answers
44 views

Subgroup of roots of unity of a field. [closed]

Let $F$ be a field. Show that the set of all $n$th roots of $1$ is a subgroup of $F^\times$.
1
vote
1answer
59 views

If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
0
votes
1answer
57 views

Prove $Z_{p}$ are prime fields,where $p$ is prime numbers

show that $Z_{p}$ are prime fields,where $p$ is prime numbers. maybe this problem is old,But I look for some book,and can't find it,someone know which book have this problem proof? because I know ...
0
votes
2answers
50 views

Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
0
votes
1answer
15 views

Splitting Field of Cubic Polynomial Over the Rationals

I'm having a hard time wrapping my head around some of concepts Pinter's Abstract Algebra introduces about splitting fields (or root fields, as it calls them). Hopefully if I can be pointed in the ...
3
votes
1answer
35 views

Embedding Fields in Matrix Rings

Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb ...
2
votes
1answer
149 views

Is $\mathbb{R}^n$ a field?

Is $\mathbb{R}^n$ a field for all $n$? I suppose for n=1 and 2 the result is clear. What about higher values of $n$.
3
votes
2answers
136 views

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
2
votes
2answers
45 views

$x\in K(t)$ is algebraic over $K$ if and only if $x\in K$ (with $t$ transcendental over $K$)

I already proved the following: Lemma A: Let $K$ be a field. Let $t$ be transcendental over $K$. Let $f \in K[X] \backslash K$. Then $f(t)$ is transcendental over $K$. Proof: Because $t$ is ...
3
votes
0answers
29 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
3
votes
1answer
50 views

Is $\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$ a simple extension?

Is the extension $$\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$$ simple? If so find the minimal polynomial and the basis for the extension.
2
votes
0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
1
vote
2answers
47 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
1
vote
0answers
43 views

Prove that every sum of squares in $K$ is a square in $K$, where $K$ is certain field.

Let $K$ be a field such that $f(t)=t^{2}+1$ is an irreducible polynomial in $K[t]$. Let $i$ be a root of $f$ in an algebraic closure of $K$. Suppose every element of $K(i)$ is a square in $K(i)$. ...
2
votes
4answers
144 views

Show that in a field always $0\ne1$

Suppose that $F$ is a field and prove that $0\ne1$ According to the definition of a field I know that the zero element is different from the one element, but is there a scientific proof for that?