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)

0
votes
1answer
53 views

If degree of extension is infinite then intermediate ring not need to be a field. [closed]

Let $F\subset K$ be a field extension and $D$ be an intermediate ring such that $F\subset D\subset K$. If $[K:F]$ is infinite then $D$ is not necessary a field. So basically I need a counter ...
1
vote
1answer
25 views

Counting elements of $\Bbb{Z}/2\Bbb{Z}(\alpha)$

I have the field $K=\Bbb{Z}/2\Bbb{Z}$, I proved that the polynomial $P(X)=X^3+X^2+1$ is irreducible. Then I know that the quotient $K[X]/P$ is a field of $8$ elements. Let now $\alpha$ be a root in ...
6
votes
2answers
68 views

A degree $4$ polynomial whose Galois group is isomorphic to $S_4$.

I am reading an article about Galois groups. The article states that: It can also be shown that for each degree $d$ there exist polynomials whose Galois group is the fully symmetric group $S_d$. ...
1
vote
1answer
81 views

The smallest positive real number — or, a field plus $\{\epsilon\}$

Suppose we want there to be a smallest positive real number $\epsilon$, so we create a new field $\mathbb{E}$ with the elements $\mathbb{R} \cup \{\epsilon\}$. $\epsilon$ should be the smallest, so ...
5
votes
1answer
59 views

Easy criteria to determine isomorphism of fields?

Let $K$ be a field and $f,g$ irreducible polynomials in $K[X]$, is there a nice iff condition for $K[X]/(f)\cong K[X]/(g)$? ($\cong$ denotes an isomorphism that is the identity on restriction to ...
0
votes
2answers
31 views

Prove that in an integral domain, if every two elements have a gcd, every irreducible element is prime

The proof of the following proposition is not completely clear to me. I get everything up until the bold part and I have a feeling some crucial steps are omitted, can anybody help clear this up? ...
2
votes
2answers
46 views

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...
10
votes
4answers
151 views

Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$. i) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ...
9
votes
5answers
781 views

The real numbers are a field extension of the rationals?

In preparing for an upcoming course in field theory I am reading a Wikipedia article on field extensions. It states that the complex numbers are a field extension of the reals. I understand this ...
1
vote
1answer
26 views

Splitting field is countable

I'm trying to prove that if $K$ is a countable field then there exists a countable field $L$ containing $K$ such that every polynomial in $K[X]$ splits in $L$. I know that if $L$ is the splitting ...
1
vote
1answer
30 views

How many distinct roots within an algebraic closure

Let $E=\overline{F_2}$. How to find the number of distinct roots of $f(x)=x^{81}-1\in F_2[x]$ in $E$? So far as I tried, I factorised $f$ into $$f(x)=(x-1)(x^{80}+x^{79}+\cdots+x+1)=(x-1)g(x)$$ ...
4
votes
1answer
30 views

Examples where $H\ne \mathrm{Aut}(E/E^H)$

If $E/F$ is a field extension, and $H$ is a subgroup of $\mathrm{Aut}(E/F)$, it is quite trivial to see that $H\subset \mathrm{Aut}(E/E^H)$. Since the theorem only shows the inclusion relationship, ...
0
votes
0answers
24 views

Extension of derivations centred in a point of an affine variety

Let be $X$ and $Y$ two affine irreducible varieties over an algebraic closed field $K$. Let be $f:X\rightarrow Y$ a surjective morphism. Then we have the immersion $f^{*}:K(Y)\rightarrow K(X)$. I know ...
1
vote
0answers
43 views

Raising to the pth power and using Vieta in finite fields with characteristic p

Any finite field $K$ with characteristic $p$ is isomorphic to $\mathbb{Z}_p[t]/(f)$ for some irreducible $f\in\mathbb{Z}_p[x]$. (From now on, assume $K=\mathbb{Z}_p[t]/(f)$.) Since ...
-1
votes
1answer
42 views

How to proof that a field is complete order field?

I know what an ordered field is but how to actually proof that a field, for example $Q$ (rational numbers), is an ordered field?
1
vote
2answers
66 views

Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$

The question as written in the title:- Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$. My attempt:- I know is $g=\gcd(f,f^{'})$ is not constant and hence ...
0
votes
1answer
41 views

Irreducible polynomial of degree $3$ and degree of extension

Let's assume we have an irreducible polynomial of degree $3$ on $\mathbb{Q}$. What are the possibility of degree of extension given by a splitting field on $\mathbb{Q}$. That is, if $K$ is splitting ...
2
votes
1answer
42 views

$k$-subalgebra of finite extension of $k$ is a field

I am working on a proof which has Let $\mathfrak m,\mathfrak n$ be maximal ideals and $A$ Noetherian. Given that, $A[T_1,\dots,T_n]/\mathfrak n$ is a finite extension of $A/\mathfrak m$, if ...
1
vote
0answers
35 views

Show that if a field is perfect any irreducible polynomial is separable.

Let $K$ a field of characteristic $p>0$. We say that a field is perfect if all algebraic extension is separable. Show that the following assertions are equivalents: 1) $K$ perfect, 2) Every ...
8
votes
1answer
123 views

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Suppose $K$ is a field and $F$ is an algebraic extension of some degree $n=(F:K)$. It is stated that the field of rational functions $F(x)$ is in fact an algebraic extension of the field $K(x)$ and ...
1
vote
3answers
73 views

What are the fields such that $x^4 = 1$ for every $x$ in the multiplicative group

What are the fields $K$ such that $x^4 = 1$ for every $x \in K^{\ast}$, i.e. such that every element of the multiplicative group is a root of $x^4 - 1$? Of course the finite fields of order $3$ and ...
3
votes
3answers
104 views

Automorphism group of an infinite field.

My primary question is: is there an infinite field $F$ with a finite automorphism group $\text{Aut}(F)$? So I tried fields with characteristic $2$, say $F_2(\pi)$. But it's still hard to make a ...
1
vote
2answers
36 views

Is the Splitting Field necessarily a subset of a field where the polynomial splits?

Just a very basic theoretical question that has been puzzling me. Let $f(x)$ be a polynomial with coefficients in a field $F$. Let $K$ be the splitting field of $f$ over $F$. Say $f$ splits in a ...
0
votes
2answers
32 views

How do we think of a field in the context of group theory?

The definition of a field (in the context of group theory) that I've been taught is as follows: "A field is defined as being a set $F$, combined with the binary operations $+$ and $\cdot$" This (to ...
5
votes
1answer
33 views

Complex subfields of finite index

It is known that the field $\mathbb{R}$ of real numbers is a complex subfield of index 2, that is, $[\mathbb{C},\mathbb{R}]=2$. Given an integer $n>2$ fixed, does there exist a subfield of ...
1
vote
3answers
68 views

Proof that $(u+v)^p = u^p + v^p$ in vector space over finite field of characteristic $p$.

Let $\mathbb F_q$ with $q = p^n$ a finite field of characteristic $p$. Then for all $x,y \in \mathbb F_q$ we have $(x+y)^p = x^p + y^p$. If $V$ is a finite-dimensional $\mathbb F_q$-vector space of ...
0
votes
0answers
51 views

$k(X)$ is not algebraically closed.

Maybe this is a stupid question but I can't find a solution: Consider an algebraically closed field $k$. Then, why the function field $k(X)$ is not algebraically closed?
0
votes
0answers
29 views

$X^p - t$ irreducible in $K[X]$ with $K:=\text{Quot}(\mathbb{F}_p[t])$

I've heard a teacher say that this also follows by the Eisenstein criterion by checking with $t$. But then $t$ would have to be prime in order for Eisenstein to have any validity, but $t$ is ...
0
votes
2answers
65 views

Annihilator of a finitely generated $K[x]$-module

Suppose $Q$ is a finitely generated $K[x]$-module for some field $K$. I want to show that $\mathrm{ann}_{K[x]}(q) = r(x)K[x]$ for some $r(x) \in K[x]$ and for all $q \in Q$. So, I already showed ...
1
vote
1answer
50 views

f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of f(x).

Consider the following theorem from book John A.Beachy Abstract Algebra. Proposition 8.6.2. I don't understand why it can be extended to an automorphism of the splitting field $F$. Can someone ...
1
vote
3answers
49 views

Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$?

Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$ ? The solution goes like : Consider $\varphi:\mathbb Z\longrightarrow K$. Since $K$ is finite $\ker \varphi=p\mathbb Z$ for a prime $p$. Q1) ...
0
votes
2answers
37 views

If $a,b\in K$ are algebraic over $k$, then $a\pm b, ab$ and $ab^{-1}$ are also algebraic.

1) Let $K/k$ a field extension. Show that if $a,b\in K$ are algebraic over $k$, then $a\pm b, ab$ and $ab^{-1}$ are also algebraic. 2) Deduce that $\{a\in K\mid a\text{ algebraic over }k\}$ is a ...
1
vote
0answers
37 views

Proof that $M \;$is a field [duplicate]

Let $\; p,d \,$ be prime numbers I wanna proof that $\; M:=\{a_o+a_1\sqrt p +a_2\sqrt d +a_3\sqrt {pd}:a_o,a_1,a_2,a_3 \in \mathbb Q \}\;$ is a field. The only thing I am having trouble with is ...
0
votes
0answers
18 views

Normal basis theorem for tamely ramified extension

Let $L/K$ be finite extension of complete discretely valued fields, $R,S$ their ring of integers, in Cassels and Frolich's Algebraic number theory p.22 line 1, they stated the normal basis theorem as: ...
1
vote
4answers
109 views

Irreducible of $\mathbb Z/p\mathbb Z$.

What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no ...
1
vote
1answer
29 views

The field $\mathbb F_q$ is it $\cong \mathbb F_p[X]/(X^q-X)$?

I know that $\mathbb F_q$ where $q=p^n$ is a field with $q$ element and that is by definition the splitting field of $X^q-X$ Q1) Is it really by definition ? Can we say that $\mathbb F_q\cong ...
2
votes
1answer
59 views

Does $\mathbb Q(\alpha )\mathbb Q(\beta )=\mathbb Q(\alpha ,\beta )$?

Does $\mathbb Q(\alpha )\mathbb Q(\beta )=\mathbb Q(\alpha ,\beta )$ ? I recall that $EF=\{ef\mid e\in E, f\in F\}$. It's clear that $\mathbb Q(\alpha )\mathbb Q(\beta )\subset \mathbb Q(\alpha ...
1
vote
2answers
45 views

Let $r\in\mathbb{Q}[\sqrt2]$. Show $\phi(r) = r$ if and only if $r\in\mathbb{Q}$

Let $\mathbb{Q}[\sqrt2]=\{a+b\sqrt2 \mid a,b\in\mathbb{Q}\}$ and define $\phi:R \rightarrow R$ by $\phi(a+b\sqrt2)=a-b\sqrt2$. Show $\phi(r) = r$ if and only if $r\in\mathbb{Q}$. My approach is ...
4
votes
1answer
61 views

Multiplication table of a Galois group?

I'm looking at the polynomial $x^4 − 4x^2 + 16$. I know that its roots are $1\pm\sqrt{3}$ and so its normal field extension is $\mathbb Q(i, \sqrt{3})$. However, I am also asked to give a ...
7
votes
2answers
43 views

Any field automorphism of $p$-adics is continuous? [duplicate]

How do I see that any field automorphism of $\mathbb{Q}_p$ is continuous?
4
votes
2answers
49 views

Question on proof about finite fields mentioned on wikipedia

In the first paragraph of wikipedia:Finite fields they write The identity $$ (x + y)^p = x^p + y^p $$ is true (for every $x$ and $y$) in a field of characteristic $p$. For every element ...
0
votes
1answer
20 views

For each $x \in GF(p^{2n})$ it is $1 - x^{p^n + 1} \in GF(p^n)$

Let $GF(p^{2n})$ be a finite field of order $p^{2n}$. Then $GF(p^n) \subseteq GF(p^{2n})$. Why do we have $1 - x^{p^n+1} \in GF(p^n)$? I know that $x^{p^n} - x = 0$ holds iff $x \in GF(p^n)$, but why ...
1
vote
3answers
39 views

Show that E is a splitting field for these polynomials

Show that if $E := \frac{\mathbb{F}_2[x]}{(x^4+x+1)}$, then $E$ is a splitting field for i) $x^4+x+1$ ii) $x^2 +x+1$ So I am trying to study for an exam and I realized that for part a, it has to ...
1
vote
2answers
44 views

Show that this field is finite and count its elements [duplicate]

I want to show that $\Bbb Z[i]/(2+3i)$ is a finite field and count it's elements. I don't really know how to show that this field is finite. I start by trying to understand that definition of this ...
4
votes
1answer
46 views

$\mathbb{Q}(\zeta)$ contains a unique subfield $K$ of degree $10$ over $\mathbb{Q}$?

Let $\zeta$ be a $151$th root of unity, $L = \mathbb{Q}(\zeta)$. How do I see that the cyclotomic field $L$ contains a unique subfield $K$ of degree $10$ over $\mathbb{Q}$? Can we conclude that ...
4
votes
1answer
37 views

$|\chi(\mathfrak{a})| = 1$ for any ideal $\mathfrak{a}$?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
1
vote
1answer
34 views

Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
-1
votes
1answer
39 views

In finite field why $\overline c = -c \ne 0$ for $\overline c = c^{p^f}$.

Let $GF(p^{2f})$ be a finite field of order $p^{2f}$. Consider the map $\overline x := x^{p^f}$ for $x \in GF(p^{2f})$. Let $b \in GF(p^{2f}) - GF(p^f)$ and set $c := b - \overline b$. Why do we ...
0
votes
1answer
20 views

When is $u_1 = w_1 + aw_2, u_2 = w_2 + a w_1$ a basis if $w_1, w_2$ is a basis.

If $V$ is a $2$-dimensional $K$-vector space with basis $w_1, w_2$, when is $$ u_1 := w_1 + a w_2 \qquad u_2 := w_2 + a w_1 $$ is basis. Of course for $a = 1$ it is certainly not, but how could ...
3
votes
0answers
28 views

The degree of a primitive element

Suppose you have a map of smooth projective irreducible curves $X \to Y$ over $\mathbb{k}$ of degree $r$. This give an extension of fields $\mathbb{k}(Y) \hookrightarrow \mathbb{k}(X)$. The primitive ...