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
39 views

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$.

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$. I tried in this way.Let $E$ be a a finite extension of $\Bbb R$. Then $E$ is an ...
2
votes
1answer
23 views

Are the following options correct in case of a field?

I am reading field theory and i can't answer the following: 1.Is $\Bbb R$ algebraic over $\Bbb Q$? 2.If a field is algebraically closed then it has characteristic as $0$. Obviously $[\Bbb R:\Bbb ...
1
vote
1answer
28 views

Can someone please explain why it is the *smallest* subfield?

I am reading field theory and having trouble with: As Fraleigh writes: Let $E$ be an extension of $F$ .Define $\phi_\alpha:F[x]\to E;\phi_\alpha(a)=a;a\in F,\phi_\alpha(x)=\alpha$ . Suppose that ...
3
votes
1answer
41 views

Field extension whose tensor product with itself over $\mathbb{Q}$ is not a field

An old qual problem reads Let $D$ be a 9-dimensional central division algebra over $\mathbb{Q}$ and $K \subset D$ be a field extension of $\mathbb{Q}$ of degree $>1$. Show that $K ...
6
votes
2answers
408 views

Does the set of all fields exist ?

We often say "let F be a field", so I was wondering if we could consider, in ZFC, the set of all fields without some contradictions arising (so that we wouldn't have to use the global axiom of choice ...
0
votes
1answer
51 views

Lang, Algebra problem

Let $k$ be a field, $f(X)$ an irreducible polynomial in $k[X]$, and let $K$ be a finite normal extension of $k$. If $g$, $h$ are monic irreducible factors of $f(X)$ in $K[X]$, show that there ...
0
votes
0answers
18 views

Irreducibility of the polynomial [duplicate]

I am trying to solve this problem from "Number Theory, Shafarevich" Any ideas? Thanks
0
votes
1answer
59 views

Prove that $F[x,y]/\langle x^2-y\rangle$ is never isomorphic to $F[x,y]/\langle x^2-y^2\rangle$, where $F$ is a field

Prove that $F[x,y]/\langle x^2-y\rangle$ is not isomorphic to $F[x,y]/\langle x^2-y^2\rangle$, where $F$ is a field. My solution. If they are indeed isomorphic, then they are isomorphic for any ...
2
votes
1answer
60 views

Is $\mathbb{C}(x,y)$ a rational function field?

Let $\mathbb{C}(x,y)$ be a degree $2$ extension of $\mathbb{C}(x)$ where $y$ is a root of $p(Z)=Z^2 + (x^2+1)$. Is it true that $\mathbb{C}(x,y)$ is not a rational function field? In other words, ...
3
votes
1answer
48 views

Show that $\beta $ is algebraic over $F(\alpha)$.

I have started reading field theory. Let $E$ be an extension field of $F$ and let $\alpha,\beta\in E$.Suppose that $\alpha $ is transcendental over $F$ but algebraic over $F(\beta)$. Show that ...
2
votes
3answers
28 views

Corollary on splitting field of polynomial

Need help in understanding a point in a proof: Let $F$ be a commutative field and $p\in F[X]$, $\deg (p) \geq 2$, an irreducible polynomial, then the ring $F[X]/pF[X]$ (quotient) is a field. $pF[X]$ ...
2
votes
1answer
48 views

roots of multi-variable polynomials and extension fields

I am teaching a course in (standard single-variable) Galois theory and the following, presumably naive, question occurred to me: Given a finite collection of polynomial equations in a finite number ...
0
votes
1answer
37 views

$[K(\zeta_n):K]=\phi(n)$?

Is it true that $[K(\zeta_n):K]=\phi(n)$ where $\zeta_n$ be primitive root of unity and K be field of char zero? I think it should be equal to the degree of cyclotomic polynomial which has degree ...
2
votes
0answers
36 views

Show that $|\text{Hom}_k(K,\widetilde{K})(\phi)|\le [K:L]$

Let $K/k$ be a finite field extension, $L$ an intermediate field and $\widetilde{K}$ such that $\widetilde{K}/k$ is normal. Let $\phi \in \text{Hom}_k(L,\widetilde{K}) := \{\psi:L\rightarrow ...
1
vote
1answer
58 views

Show that the only subfields of $\mathbb{Q}(i, \sqrt{5})$ is $\mathbb{Q}, \mathbb{Q}(i),\mathbb{Q}(\sqrt{5}), \mathbb{Q}(i \sqrt{5})$ and itself?

I'm reading Stewart's Galois Theory and encountered this exercise in Chapter 8. I want to show this by contradiction: Assuming there exists a proper subfield $\mathbb{Q}(\alpha)$ of $\mathbb{Q}(i, ...
4
votes
3answers
198 views

Rings that are generated as an Algebra over a field by an arbitrary amount of algebraic elements

In an introduction course in algebra, you learn, that if you take a field $F$ and an element $x$, which is algebraic over $F $, then the smallest generated Ring by $F$ and $x$, mostly called $F[x]$ is ...
1
vote
0answers
25 views

Which of the following field properties are correct?

Let $\omega = \cos{\frac{2\pi}{10}}+i\sin{\frac{2\pi}{10}}$. Let $K = \mathbb{Q}(\omega^2)$ and $L = \mathbb{Q}(\omega)$. Then $[L : \mathbb{Q}] = 10.$ $[L : K] = 2$. $[K : ...
0
votes
2answers
64 views

If b is algebraic over F(a), and a is algebraic over F, then is b algebraic over F?

If $b$ is algebraic over the field $F(a)$ then is it algebraic over the field F? I would like to find a proof if it is true or a counter example if it is not. The only thing I could think of was ...
3
votes
0answers
34 views

Smallest intermediate field containing two intermediate fields

Let $K/k$ be a field extension and $A$ and $B$ intermediate fields. Further, let $C$ be the smallest intermediate field that contains $A$ and $B$. (1) Show that $A/k$ and $B/k$ separable $\implies$ ...
-1
votes
1answer
15 views

find all composite order fields between 200 and 900 . [closed]

There are many fields of composite order between 200 and 900 how can I find those fields.
2
votes
1answer
81 views

The maximal subfield of $\mathbb C$ not containing $\sqrt2$

Related: Does a maximal subfield of $\mathbb C$ not containing $\sqrt{2}$ have index $2$? He said, "...fixed field is an extension of $K$ which doesn't contain $\sqrt{2}$, and thus must be $K$ ...
0
votes
1answer
17 views

The Galois group of automorphisms on the splitting field of the polynomial x^5 - 11.

I think that the splitting field (the smallest subfield of C that contains all the roots of x^5 - 11) is Q adjoined with r and z where r is the real solution of x^5 - 11 = 0 and z is the 5th root of ...
7
votes
1answer
78 views

Embedding fields into the complex numbers $\mathbb{C}$.

Let $k$ be a field of characteristic $0$ with $\mathrm{trdeg}_\mathbb{Q}(k)$ at most the cardinality of the continuum. I want to prove the existence of a field homomorphism $k\rightarrow\mathbb{C}$. ...
2
votes
0answers
37 views

possible degrees of the minimal polynomial of t over F [closed]

Please help me with this Abstract Algebra question: Let $F \subset K$ be a Galois extension with Galois group isomorphic to the alternating group $A_4$. Let $t \in K$ be an element. Determine the ...
3
votes
2answers
65 views

Determine whether the splitting field of a polynomial contains a subfield M such that M:$\mathbb {Q}$ is not normal

For the following polynomials I need to find out if the splitting field over $\mathbb {Q}$ has a subfield M such that M:$\mathbb {Q}$ is not normal. 1) $ x^6-7$ 2)$ x^3 + 3x +3 $ 3)$x^{100} - 1$ ...
4
votes
1answer
104 views

A Galois theory sanity check about conjugates.

Here is my question... If $L/K$ is an algebraic extension and $\alpha,\beta \in L$ are $K$-conjugates (that is, they have the same minimal polynomial), is it always true that there exists some ...
1
vote
1answer
28 views

Minimal polynomial of an element of a field is a minimal polynomial of a matrix?

Let $F$ be a field and $K/F$ be a finite extension. For any $x \in K$, there is a minimal polynomial for $x$. On the other hand, the multiplication by $x$ induces a $F$-linear map $K \to K$. This ...
0
votes
1answer
20 views

$K(ab,a+b) \subset K(a,b)\;$ finite field extension

Let $\; K(ab,a+b) \subset K(a,b)\subset L \quad a,b \in L$ Is $\; K(ab,a+b) \subset K(a,b) \;$ a finite field extension and if not can anyone give a counterexample ?
3
votes
2answers
43 views

Degree of the splitting field of $ x^3-5 $ over $\mathbb{Q}$

I am trying to find the degree of the splitting field for $ x^3-5 $ over $\mathbb{Q}$. I have so far: The splitting field will be $\mathbb{Q}(\sqrt[3]{5},u)$ where u is the 3rd root of unity. So ...
0
votes
3answers
60 views

Prove that $\sqrt{2} \notin \mathbb{Q}(\sqrt{3})$.

Suppose there exists an isomorphism $\Phi \colon \mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{3})$. Then, of course, it must be the case that $\Phi(1) = 1$. Hence \begin{align*} 2 &= 1+1 = \Phi(1) + ...
2
votes
2answers
55 views

Primitive Element theorem, permutations

Let $K = \mathbb{Q}(\alpha_1,\alpha_2,...\alpha_n)$, where the $\alpha_i$ are the roots of some irreducible polynomial (and hence they are pairwaise distinct since the polynomial is separable). Then ...
0
votes
1answer
23 views

Is a finite simple extension of fields in characteristic zero already normal?

Let $k(\alpha) / k$ be a finite separable simple extension, $char(k) = 0$. Is $k(\alpha) / k$ already a normal extension? I can't come up with a counterexample or a proof that it is normal.
0
votes
2answers
29 views

Why in a field of characteristic $p$, $\zeta_p \sqrt[p]t$ is not a root of $X^p-t\in \mathbb F_p(t)[X]$

I know that in a field such that $Car(K)=p$ is prime, the $X^p-t\in \mathbb F_p(t)[X]$ has a unique root (I know how to prove it and thus, it's not the question). But in the usual logic, $X^p=t\iff ...
0
votes
0answers
26 views

Subfield of $\mathbb C$ with certain degree.

Let $n\in\mathbb N$. then, is there a subfield of $\mathbb C$ such that $[\mathbb C:K]$= $n$ ? if $n$ = 1,2, then answer is yes. But I don't know if $n$ is larger than 2. How do I proceed more? ...
2
votes
0answers
29 views

Galois extension of $\mathbb C$ is also Galois over $\mathbb R$??

Is any Galois extension of $\mathbb C$ also Galois over $\mathbb R$? I know if that extension is finite, Then it is true because $\mathbb C$ is algebraically closed. But How about the infinite case? ...
1
vote
2answers
18 views

If $R/J(R)$ is simple then $R$ is local

The question is what I said in the title :- If $R/J(R)$ is simple then $R$ is local. where $J(R)$ is Jacobson radical. I only have the idea about the converse ( If $R$ is is local then ...
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
30 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
778 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 ...