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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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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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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26 views

Show that the characteristic of a field is $0$ or prime. [duplicate]

Let $K$ be a field and $1_{K}$ on element of the field. We define the characteristic $char(K)$ of $K$ as $$ char(K):=\begin{cases} \text{the least number $n\geq 1$ with $n\cdot1_K=0$} ,& \text ...
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1answer
21 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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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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22 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 $n = 1$, $(\mathbb F_p,+)$ ...
3
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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 ...
2
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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. ...
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0answers
33 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
22 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
17 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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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 ...
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1answer
59 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
19 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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20 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
46 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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0answers
25 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
31 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
43 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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44 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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2answers
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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2answers
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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+200

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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3answers
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 ...
2
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1answer
35 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
468 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 ...
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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 ...
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1answer
7 views

Tensor product of fields and its subalgebra

In Nathan Jacobson's Basic Algebra II, in section 8.18: Tensor product of fields he is discussing what happens to $E \otimes_FK$, when $K|F$ and $E|F$, and E is algebraic over F. At one point he ...
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3answers
244 views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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1answer
48 views

If $k[X]/f = k[X]/g$, does $f = g$?

Let $k$ be a field and $f, g$ be irreducible monic polynomials in $k[X]$. Suppose $k[X]/f \stackrel{\sim}{=} k[X]/g$. Then does $f = g$? If so, how can this be generalized? Otherwise, how should I ...
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0answers
9 views

Norm in field theory

Let $K/k$ be a finite separable extension, and $\sigma_1,\ldots,\sigma_n$ the embedings. For each $\alpha\in K$, the $\textbf{norm}$ is $$Nr(\alpha)=\sigma_1(\alpha)\cdots\sigma_n(\alpha)$$ Then ...
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1answer
59 views

Addition in Field.

Find counterexamples to the following statements: In every field $\Bbb F$, if $a\in \Bbb F$, $a+a=0$, then $a=0$; Counterexample: Consider $\Bbb Z_2$. Let $a = 1$, so $a + a = 2 = 0 \mod 2$. ...
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2answers
44 views

Prove the fractional field of an integral domain is the smallest field containing the integral domain

I have two questions about the fractional field of an integral domain. Given an integral domain $D$: Is there a difference between saying "the fractional field of $D$ is the smallest field ...
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0answers
37 views

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
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2answers
21 views

element that is algebraic over a finite field

Let $p$ be a prime. And let $q = p^{2h}$. Suppose I know that an element $\alpha \in \overline{ \mathbb{F}_q }$, satisfies $\alpha^2 + \alpha + 1 = 0$. Does this mean that $\alpha \in \mathbb{F}_{p^2} ...
3
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1answer
42 views

Has anyone defined a limit of a sequence of fields? In particular, what is the limit of finite fields?

I'm curious about $$ \lim_{n \rightarrow \infty} \mathbb{F}_n $$ Is it $\mathbb{Z}$? That seems reasonable if you consider it as a set but of course $\mathbb{Z}$ is not a field so that is confusing. ...