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 ...

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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 ...
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18 views

Fixed field of the subgroup of $Aut_{K}{K(x)}$

This link http://www.math.umd.edu/~jmr/601/functfield.pdf explains $Aut_{K}{K(x)}$. And I want to know how to solve two problems below in the hungerford's algebra textbook, ...
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2answers
19 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 ...
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1answer
28 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 ...
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1answer
24 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 $K$ ...
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1answer
28 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 ...
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33 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 ...
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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.
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1answer
45 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 ...
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22 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 ...
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38 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 ...
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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?
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3answers
38 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 ...
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1answer
23 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 ( ...
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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 $4\cdot 4^{-1}+4^{-1}=4^{-1}(1+4)$ ...
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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 ...
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23 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 ...
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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 ...
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1answer
30 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 $n = 1$, $(\mathbb F_p,+)$ ...
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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 ...
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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. ...
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34 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: $x=[1,1,0,1]$ ...
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2answers
26 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 ...
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1answer
30 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 ...
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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 ...
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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 ...
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32 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 ...
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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 ...
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1answer
60 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 ...
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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 ...
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21 views

the number of the irreducible polynomials

i'm solving this problem the number of intermediate fields in a simple field extension of degree $n$ and i need to know if $f(x)$ has degree $n$,how many irreducible polynomials such as $g(x)$ can be ...
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3answers
47 views

Question about the Characteristic of $\mathbb{F}_{p^n}$

We can prove that any finite field of prime characteristic $p$ must have $p^n$ elements. Conversely, let $F$ be a finite field with $p^n$ elements, where $p$ is a prime number. Is the following ...
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29 views

checking the solution of problem in field extensions

here's the problem: suppose that $K|F$ is a field extension and $a\in K$ is an algebraic element over $F$ that it's minimal polynomial has odd degree.prove that $F(a)=F(a^2)$. i think this is a ...
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1answer
32 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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1answer
45 views

Characteristic 3 analogue of nimbers?

Finite nimbers are a way of turning the natural numbers (finite ordinals) into a characteristic 2 field. Addition in this field is found by writing the numbers in binary and adding without carry, ...
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12 views

Is a field a PID? [duplicate]

How can I prove that a field is a PID? I can prove that a field is an Integral Domain, but stuck in proving that every ideal is principal.
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46 views

Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
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44 views

How to prove a finite field is not ordered?

I have a set S={0,1}, and the addition and multiplication rules are \begin{array}{c|cc} +&0&1\\ \hline 0&0&1\\ 1&1&0 \end{array} \begin{array}{c|cc} *&0&1\\ \hline ...
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2answers
41 views

Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
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27 views

Galois subextensions in a Galois extension

Let $F \subset E \subset L$ be fields such that $L/E$ and $E/F$ are both Galois extensions. Is $L/F$ necessarily a Galois extension?
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193 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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0answers
35 views

Smallest field containing $\mathbb{Q}$ and closed under square root

I'm following Isaacs' Algebra and I need to prove that the field $K$ of constructible numbers is the smallest subfield of $\mathbb{C}$ such that is closed under taking square root. I already know ...
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27 views

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

I thought that surely these two fields are in fact equal. I was told otherwise by someone today. Am I just confused?
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2answers
30 views

Field extensions and irreducibility

I'm having trouble trying to show that the function f=x^3 + x + 3 is irreducible in the rationals. I tried using Eisensteins criterion but it didn't work as it doesnt satisfy all conditions. the ...
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84 views

Can $\mathbb{Z}$ be endowed with operations that give it the structure of a field?

Does there exist some definition of addition and multiplication for which the set of all integers is a field?
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24 views

Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that ...
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1answer
41 views

Non-Galois number fields and complex embeddings

Let $K$ be a number field. $K$ is a normal extension of $\mathbb{Q}$ iff $\exists f(x)\in\mathbb{Q}[x]: K$ is the splitting field for $f(x)$. A field extension is Galois iff it is normal and ...
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1answer
29 views

What is wrong with my proof of a step in Artin's construction of algebraic closure?

I'm working through Atiyah & MacDonald, and there's an exercise basically asking you to fill in a certain step in Artin's construction of an algebraic closure for a given field. The question is ...
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1answer
471 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
2
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1answer
30 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...