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

1
vote
0answers
42 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
31 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
35 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
24 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
19 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
36 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
60 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
64 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
26 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 ...
2
votes
3answers
51 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 ...
1
vote
1answer
36 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 ...
3
votes
1answer
53 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, ...
0
votes
0answers
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.
0
votes
0answers
56 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 $D$ ...
3
votes
2answers
49 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 ...
4
votes
2answers
45 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 ...
1
vote
2answers
29 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?
8
votes
0answers
212 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 ...
2
votes
0answers
42 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 ...
0
votes
0answers
29 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?
0
votes
2answers
33 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 ...
0
votes
3answers
86 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?
2
votes
0answers
27 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
votes
1answer
45 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 ...
1
vote
1answer
32 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 ...
10
votes
1answer
487 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
votes
1answer
36 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 ...
0
votes
1answer
8 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 ...
3
votes
3answers
300 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 ...
1
vote
1answer
49 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 ...
0
votes
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 ...
1
vote
1answer
61 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 $a + a = 2 = 0 \mod 2$. ...
1
vote
2answers
57 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 ...
1
vote
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 ...
2
votes
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
votes
1answer
46 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. ...
2
votes
0answers
40 views

Brauer groups of curves and base change

Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements ...
2
votes
1answer
34 views

relation between the characteristic polynomial and the minimal polynomial

Define $l_a : F(a) → F(a) $ by $ l_a(x)=ax$, when $[F(a):F]=n$ . show that the minimal polynomial of $a$ over $F$ is the same as the minimum polynomial of $l_a$ as defined in linear algebra. this ...
1
vote
2answers
35 views

Suppose that $L(\alpha):L:K$ and that $[K(\alpha):K]$ and $[L:K]$ are relatively prime.

Show that the minimal polynomial of $\alpha$ over $L$ has its coefficients in $K$. I tried an approach but I got stuck: We have that the following field extensions: $L(\alpha)/L$ and $L/K$ and we ...
1
vote
1answer
43 views

Extension of an Isomorphism

Suppose $E_1, E_2 \subset E$ are proper subfields. In general, if one has an isomorphism $\sigma:E_1\to E_2$, is it possible to extend it to an isomorphism $\psi:E\to E$ s.t. $\psi|_{E_1} = \sigma$ ...
0
votes
2answers
50 views

Prove all elements of $A$ is algebraic over $C$, if all elements of $A$ are algebraic over $B$ and $B$ are algebraic over $C$

Let there be 3 fields $A$, $B$ and $C$. If all elements of $A$ are algebraic over $B$ and all elements of $B$ are algebraic over $C$, prove that this implies that all elements of $A$ is algebraic ...
1
vote
1answer
25 views

a question about field extensions and tower formula

if $K|F$ is a field extension & $a_1,a_2,...,a_n$ are the elements of $K$ which are algebraic on $F$ , we know that $[F(a_1,a_2,...,a_n):F]=<\Pi_{i=1}^n[F(a_i):F]$,it can be proved by induction ...
0
votes
1answer
41 views

Irreducible polynomial iff the condition is satisfied

I am asked to show that the polynomial $f(x)=x^n+1 \in \mathbb{Q}[x]$ is irreducible if and only if $n=2^k$ for any integer $k\geq 0$. Could you give me some hints what I could do to show this??
0
votes
1answer
11 views

Normal closures of transcendental extensions

If $E$ is a finite algebraic extension of the field $F$, then we can find a normal closure of $E$ over $F$. What can we say if the extension $E$ is not finite?
4
votes
1answer
34 views

Fields extensions over isomorphic fields of different degrees

What are the simplest examples of situations where in a field $F$ there are two subfields $L_1$ and $L_2$ such that extensions $F/L_1$ and $F/L_2$ are finite, degrees are different $$ [F:L_1] \neq ...
0
votes
0answers
48 views

The function field of $V=Z(y^2-x^3)$

Let $k$ be a field and let $V=Z(y^2-x^3).$ Can someone explain to me why $k(V)\cong k(s,t)$ ?? with $t=x+(y^2-x^3),s=y+(y^2-x^3)\in A(V)=k[x,y]/(y^2-x^3).$ Can we generalize it : If $V=Z(f)$ with ...
0
votes
1answer
53 views

Show that it is a field

$K \leq E$ an algebraic extension. I am asked to show that each subring of $E$ that contains $K$ is a field. I have done the following: $K \leq E$ algebraic $\Rightarrow \forall a \in E, \exists ...
4
votes
1answer
78 views

On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
0
votes
2answers
37 views

Field extension-degree

I have the following question... $K\leq E$ a field extension. When we have that $$[E:K]=1$$ do we conclude that $K=E$?? Or must also something else be satisfied so that $K=E$ ??
0
votes
2answers
46 views

Field extension-Why does this hold?

$K\leq E$ a field extension, $a\in E$ is algebraic over $K$. Could you explain me why the following holds?? $$K\leq K(a^2)\leq K(a)$$