Tagged Questions

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

learn more… | top users | synonyms

0
votes
0answers
4 views

a problem about normal extensions & automorphisms

this is my problem: Suppose $K|F$ is a normal extension,prove that for every $\alpha ,\beta \in K$ that have the same minimal polynomial over $F$,there is a F-algebra automorphism over ...
1
vote
2answers
19 views

Maximal ideal in $\mathbb{Q}[x,y]$

I am trying to prove that $(x,y)$ is a maximal ideal of $\mathbb{Q}[x,y]$. Since an ideal $I \subseteq R$ is maximal if and only if $R/I$ is a field, it suffices to prove that $\mathbb{Q}[x,y]/(x,y)$ ...
0
votes
2answers
18 views

a problem about splitting field & irreducibility of a polynomial

suppose that $K$ is the splitting field of $f(x)\in F[x]$ ,when the degree of $f(x)$ is $n$ & $[K:F]=n!$.show that $f(x)$ is irreducible over $F$. i know that $K|F$ is normal,but i don't know how ...
2
votes
0answers
21 views

Splitting field in finite field

What is the splitting field of the polynomial $X^{p^8}-1$ over $\mathbf F_p$? I'm confused, not is $X^{p^8}-1=(X-1)^{p^8}$ then the splitting field is $\mathbf F_p$? Thanks.
0
votes
1answer
39 views

Suppose you are given an ordered field $F$. You dont know exactly what set $F$ is, but…

Suppose you are given an ordered field $F$. You dont know exactly what set $F$ is, but you know there exists a nonempty subset $A\subset F$ with no upper bound. What can we say about $F$? Namely, can ...
3
votes
3answers
70 views

Proving two finite fields are isomorphic

So I'm asked to prove that $\mathbb{F}_9$, defined as $\{ a+bi$ | $a,b \in \mathbb{Z}_3,$ $i^2 = 2 \}$, is isomorphic to the field $F_1$, defined as $\mathbb{Z}_3[x]/ \langle x^2+2x+2 \rangle$, where ...
0
votes
0answers
7 views

Can cancellation property of finite vector space be used to show uniqueness of vectors combination?

Here are several questions together, but they need to be related (hopefully). If set of vectors is a basis, then, considered with all its linear combinations it forms a field (with vector addition ...
0
votes
2answers
36 views

Does every infinite field contain the integers as a subring?

I simply ask because if $1+1=2(1)=2$ then this would imply that all positive integers are contained, and as every element in a field has a negative all the negative integers are contained. At the same ...
0
votes
0answers
33 views

existence and meaning of inverse of an element in Q(α)

I can't understand when exist the inverse of an element in Q(a). For example I read an example $f = 3 + 2*x$ and $a = sqrt(2)$ and computes the inverse with extended euclidean algorithm, but I am a ...
0
votes
0answers
21 views

Basis for a field extension

Suppose that $K=\mathbb Q(\alpha)$ is normal and $\alpha,\alpha_2,\ldots,\alpha_n$ the conjugates of $\alpha$. Is necessarily $\{\alpha,\alpha_2,\ldots,\alpha_n\}$ a basis for $K$ over $\mathbb Q$? ...
0
votes
0answers
31 views
+50

a problem about normal extensions & polynomials

Suppose that $K|F$ is a normal extension.prove that for every irreducible $f(x)\in F[x]$,every irreducible polynomial such as $g(x) \in K[x]$ that $g(x)|f(x)$ have the same degree. i started to prove ...
2
votes
2answers
49 views

How to find a minimal polynomial

I need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?
2
votes
2answers
17 views

Example of a normal extension.

Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the ...
0
votes
0answers
10 views

Norm in cyclotomic field

Suppose $p$ is a rational prime and $\zeta=e^{2\pi i/ p}$. Prove that the groupp of non-zero elements of $\mathbb Z_p$ is cyclic, show that there exists a monomorphism $\sigma:\mathbb Q(\zeta)\to ...
0
votes
0answers
25 views

Extension fields3 [on hold]

If $F/K$ is an extension and $a,b \in F$ algebraic over $K$ of degree $n$ and $m$ respectively. Prove that: $[K(a,b):K]\leq nm$; If $n$ and $m$ are relatively prime, then $[K(a,b):K]=nm$; If $n$ and ...
2
votes
0answers
17 views

an exercise about normal extensions

is the extension $\mathbb {Q}(2^\frac{1}{2},2^\frac{1}{3})$ over $\mathbb{Q}$ normal? i think not,because the polynomial $x^3-2$ over this field has just one root:$2^\frac{1}{3}$ ,and other roots ...
1
vote
0answers
34 views

Proof about field extension : A geometric way

Let $M \subset \mathbb C $ be a sub-field which is not contained in $\mathbb R$ and which is closed under complex-conjugation. Let $L(M)$ be the set of all lines which crosses two points of $M$ and ...
-1
votes
0answers
23 views

can't understand a part of Lorenz's Galios theory book [on hold]

i can't understand this part from the book of Lorenz,Fields and Galios theory. page 60-63,from where it says: We now equip ourselves with an important algebraic tool, which we will use to prove ...
0
votes
1answer
19 views

Transcendental extension over a field K.

Prove that $x$ is transcendental over $F(x)$ or more generally show that any element not in $K$ but in $K(x_1,x_2,x_3,x_4,\ldots,x_n)$ is transcendental?
2
votes
2answers
34 views

Extension field, degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$

I want to calculate the degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$, can I do like that: $$X=i+\sqrt{-3}\implies X=i(1+\sqrt{3})\implies X^2=-(1+\sqrt{3})^2\implies X^2=-1-2\sqrt{3}-3\implies ...
0
votes
0answers
25 views

Roots Of An Inseparable Polynomial.

Suppose that we have a field $K = \mathbb{Z}_p(t)$ where $p$ is prime. (So this is the field of rational polynomials with $t$ as the variable) Let $f = x^p - t$ be a polynomial in $K[x]$. How can ...
2
votes
1answer
31 views

Fixed field of two subgroups of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra, p.256. $7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ ...
1
vote
2answers
22 views

Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
1
vote
1answer
32 views

linear independence and field extension

suppose $K|F$ is a field extension & $\alpha \in K$ is such that $[F(\alpha):F]>=n$, if $\lambda_1,...,\lambda_n$ are distinct scalars of $F$,prove that ...
1
vote
1answer
40 views

Suppose that $L:K$ is algebraic. Show that the following are equivalent:

$(A)$ $L:K$ is normal $(B)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ then $j(L) \subseteq L$ $(C)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ ...
3
votes
1answer
30 views

The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds. How can I use ...
4
votes
1answer
34 views

Suppose $F$ is a finite field of characteristic $p$. Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$.

Suppose $F$ is a finite field of characteristic $p$ ($p$ a prime). Prove $\exists u \in F$ such that $F = \mathbb{F}_{p}(u)$. Here, $\mathbb{F}_{p}$ denotes the field with $p$ elements. Here is ...
0
votes
1answer
30 views

field of fractions and being algebraically closed

prove that for every field $F$ the field of fractions $F(x)$ is not algebraically closed. it is a problem which i don't know how to deal with it. help please. thank you.
1
vote
1answer
54 views

$f(x)$ is still irreducible

Let $f(x) \in K[x]$ an irreducible polynomial of $K[x]$ of degree $n$. Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible also as a polynomial of ...
2
votes
2answers
23 views

monomorphism on an algebraic field extension

let $E|K$ be an algebraic extension and $\phi:E\rightarrow E$ a $K $-algebra monomorphism,prove that $\phi$ is onto. i assume $\alpha\in E-\phi(E)$ to make a contradiction and i assume $f(x)$ to be ...
1
vote
0answers
40 views

Diagrams characterizing ring characteristic, and in particular field characteristic 1?

For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$? Is there a diagram, or the shape of what ...
0
votes
2answers
9 views

Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_i=c_j$ in a field?

Does $\sum_{i=1}^nx_i=0$ and $\sum_{i=1}^nc_ix_i=0$ implies $c_1=\cdots=c_n$ where $c_i$ and $x_i$ are elements of a field $F$? If so, why?
1
vote
3answers
39 views

What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R ...
2
votes
1answer
26 views

Show that it is a subfield

To show that $$\mathbb{Q}(2^{1/x}, 2^{1/y}) \subseteq \mathbb{Q}(2^{1/{xy}})$$ knowing that $(x,y)=1$, $x, y \in \mathbb{N}$ can we do the following?? $$2^{\frac{1}{x}}=2^{\frac{y}{xy}}=\left ( ...
0
votes
1answer
19 views

Problem with modulo in field

I have problem with comprehending how works number in field when it's rasied to negative power. For instance if we have $4^{-1}$ at $Z_{5}$ I tried to write it as $4\cdot 4^{-1}+4^{-1}=4^{-1}(1+4)$ ...
0
votes
0answers
15 views

Why does this criterion imply that $A$ is a subfield of $E$?

$E$ is an extension field of a field $F$ and $A$ is the subset of $E$ containing all the members algebraic over $F$. "To prove that $A$ is a subfield of $E$ it is enough to show that any two elements ...
1
vote
0answers
24 views

Algebraic extension of rational functions

Let $k\subset F\subseteq k(X)$ be chains of field extension, prove that $k(X)/F$ is algebraic. "Proof:" Let $y\in F\setminus k$ then $y=\frac{P(X)}{Q(X)}$ with $P\notin k$ or $Q\notin k$. It ...
0
votes
0answers
7 views

A question from calculus my test(Curl, guess theorem )

the value of the integral $$ \iint rotF*n*ds \quad where \quad s-> x^2+y^2+z^2=4 \quad $$ and the normal is making a blunt angle with the Z axis, and $$ f=(zsinx-2y+1)i+(3x)j+(4xz+z^3)k $$ im ...
0
votes
2answers
41 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
3
votes
1answer
16 views

different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...
2
votes
0answers
30 views

Find the ring of algebraic integers. [duplicate]

Find the ring of algebraic integers in $K=\mathbb Q(\sqrt[3]{2})$. So, I know that $K=\{a+b\sqrt[3]{2}+c\sqrt[3]{2}^2 \mid a,b,c \in \mathbb Q\}$. My professor has done very little on this topic. ...
1
vote
0answers
36 views

Solving system of equation over distinct finite fields

Is it possible to solve an equation systems which involves elements from distinct finite fields? One example could be elements from $\mathrm{GF}(2)$ and $\mathrm{GF}(2^2)$: With: $x=[1,1,0,1]$ ...
0
votes
2answers
28 views

Let K be field and L be a subfield prove that

Let $K$ be field and $L$ be a subfield prove that A) if $K= \mathbb{R}$ and $\sqrt{2} \in L$ then $\mathbb{Q}(\sqrt2) \subset L$ B) If $K= \mathbb{Z}_p$ then $L=\mathbb{Z}_p$ I don't know how ...
2
votes
1answer
32 views

Definition of field of fractions of $p$-adic integers

I am given this definition of the field of fractions of the p-adic integers: $$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$ How can I show that: $Q_p$ consists of the sums of the ...
0
votes
0answers
23 views

If $k[a_1,a_2,…,a_n]=k(a_1,a_2,…,a_n)$ show that $a_1,…,a_n$ are algebraic over $k$.

I am trying to prove the following statment and need some help. Let $k$ and $E$ be fields such that $k \subset E$ and $a_1,a_2, \ldots ,a_n \in E$, if $k[a_1,a_2,...,a_n]=k(a_1,a_2,...,a_n)$ show ...
1
vote
1answer
18 views

checking a solution to an exercise in field extension

this is the exercise: suppose $K|F$ is a field extension , $\alpha,\beta\in K^∗$ , $m,n$ are two integers that $(m,n)=1$ and $α^m,β^n∈F$,prove that $αβ$ is a primitive element of $F(α,β)|F$. this is ...
1
vote
0answers
33 views

Perron-Frobenius theorem for real closed fields via model theory

The Perron-Frobenius theorem states that any matrix over the reals with positive entries has at least one positive eigenvalue (and a bit more). The easiest proof that I know of runs as follows: any ...
0
votes
0answers
58 views

My answer to this simple question is wrong but I don't know why

I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks: "Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a ...
2
votes
1answer
62 views

Showing $\operatorname{Aut}(\mathbb{R})$ is abelian

I have an exercise (not for a class) that asks whether $\operatorname{Aut}(\mathbb{R})$ (field automorphisms) is abelian. I know that $\operatorname{Aut}(\mathbb{R})$ is just the trivial group, but ...
1
vote
1answer
20 views

the number of intermediate fields in a simple field extension of degree $n$

suppose that $K|F$ is a simple field extension with degree $n$,prove that the number of intermediate fields is less or equal $2^{n-1}$. i've done this: assume $K=F(a)$ and $L$ is a intermediate ...