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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Let $F$ a field and $F^{alg}$ it algebraic closure. If $E/F$ is algebraic, does $E^{alg}=F^{alg}$ or not?

Let $F$ a field and $F^{alg}$ it algebraic closure. If $E/F$ is algebraic, does $E^{alg}=F^{alg}$ or not ? I would say yes, since every polynomial on $E$ split over $K^{alg}$ (since $E\subset ...
0
votes
1answer
32 views

Why the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ s.t. $x\longmapsto x^p$ is surjective?

Consider the frobenius $\mathbb F_p^{alg}\longrightarrow \mathbb F_p^{alg}$ defined by $x\longmapsto x^p$. 1) Why is it surjective ? I recall that $\mathbb F_p^{alg}$ is an algebraic closure of ...
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1answer
43 views

Why does an algebraically closed field not have any non-trivial algebraic field extensions?

Let $K$ be an algebraically closed field. Then there are no non-trivial algebraic field extensions of $K$. I can understand that if the field extension is of the form $K[x]/\langle p(x)\rangle$, ...
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2answers
35 views

A function in the integers module $p$ is polynomial. [closed]

Let $p$ a prime number and $\mathbb{F}_p$ the field of integers module $p$. Show that if $f:\mathbb{F}_p\to \mathbb{F}_p$ is a function then $f$ is polynomial.
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2answers
37 views

Why is every finitely generated field over $k$ not a finite type $k$-algebra?

A field extension $k\subseteq F$ is finitely generated if there exist $\alpha_a,\alpha_2,\dots,\alpha_n\in F$ such that $$F=k(\alpha_1)(\alpha_2)\dots(\alpha_n)$$ This is not the same as saying $F$ ...
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0answers
50 views

The Galois Field for the polynomial $x^3 - 2$.

I am reading a textbook prior to taking my first course in Field Theory. I think that if someone could answer the following 4 questions simply as True of False I might be less confused. I am denoting ...
3
votes
1answer
34 views

Show that certain matrices over rings form a field

I have got the following assignment: $R$ is a ring, $K:=\{ \begin{pmatrix} a & b \\ -b & a \\ \end{pmatrix}: a,b \in R\}$ I need to show that $K$ is a field. And I believe it is not ...
2
votes
1answer
122 views

Example of a chain without a supremum in a non Archimedean ordered field

I give here the example of a non-Archimedean ordered field. I know that the field is not order complete. What is a simple example in that field of a chain without a supremum?
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0answers
94 views

Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
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1answer
36 views

Finding Galois conjugates

I'm working on a big exercise from Dummit & Foote (p.584) with the end goal of constructing a polynomial with Galois group $Q_8$ (Quaternion group of order $8$). Take $$\alpha = ...
4
votes
4answers
136 views

The equation $-1 = x^2 + y^2$ in finite fields

In an ordered field we have $x^2 \ge 0$, hence the equation $-1 = x^2 + y^2$ has no solution. But what about finite fields in general? What is the solutions set $$ -1 = x^2 + y^2 $$ of this equation? ...
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votes
2answers
27 views

Splitting field of $x^2 +1 \;$ over $\mathbb Z_7 $

I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $. The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is ...
0
votes
2answers
49 views

Finite Fields problem [closed]

"Given a Galois Field $(\mathbb{F}, +, \cdot)$ of order 8. With an element $x \in \mathbb{F}$ we create a group $(\{x^m | m \in \mathbb{Z}\}, \cdot)$. ($x^m$ is calculated via the second operator ...
2
votes
1answer
28 views

Product of degree of two field extensions of prime degree

Let $L/K$ be a field extension. Let be $\alpha, \beta \in \mathbb{C}$, such that $[\mathbb{Q}(\alpha):\mathbb{Q}] = p$, and $[\mathbb{Q}(\beta):\mathbb{Q}] = q$, for some prime numbers $p$ and $q$. ...
1
vote
1answer
31 views

Galois group of a polynomial over $\mathbb{C}[t]$

To find the Galois group of the polynomial $X^3-X-t\in\mathbb{C}[t]$, an approach is to compute the discriminant (equal $(2-\sqrt{27}t)(2+\sqrt{27}t)$) which is not a square in $\mathbb{C}[t]$ so the ...
0
votes
1answer
28 views

Invariant subfields and Galois group

Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Let $\alpha\in K$ be a root of $f$. Let $H$ be the subgroup of $G$ that ...
1
vote
1answer
45 views

Intersection of Kummer extension [closed]

Let $p$ and $q$ be two prime numbers and $\omega$ be the primitive 3rd root of unity. The splitting field of $X^3-p$ over $\mathbb{Q}$ is $K_p=\mathbb{Q}(p^{\frac{1}{3}},\omega)$ and we have a similar ...
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1answer
45 views

A Question on Integral domains and Fields [closed]

Suppose $F$ is just a non-zero commutative ring with a unit. I want to ask can we deduce that $F$ is an integral domain.
0
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3answers
21 views

In the finite field $F$ of characteristic $p$, is $a^{p^n} = a$?

If F is a finite field of characteristic $p$, $a$ is some element in $F$ and the number of elements in $F$ is $p^n$, is it true that $a^{p^n} = a$ for all $a$ in $F$? If it is, how could one prove or ...
0
votes
2answers
30 views

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$.

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$. Suppose that $o(F)=p^n$ .Let $F$ has $\Bbb Z_p$ as its prime subfield. Let $n=km$. I ...
0
votes
1answer
20 views

Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ .

Consider the polynomials $x^3+x^2+1,x^3+x+1$ over $\Bbb Z_2$ which have roots say $\alpha,\beta $ respectively. Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ . Since both $x^3+x^2+1,x^3+x+1$ are ...
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
29 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
409 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 ...
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votes
0answers
18 views

Irreducibility of the polynomial [duplicate]

I am trying to solve this problem from "Number Theory, Shafarevich" Any ideas? Thanks
0
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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
64 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
203 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
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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 $\implies$ ...
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1answer
16 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
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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
81 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 $k\rightarrow\mathbb{C}$. ...
3
votes
2answers
68 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
21 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
44 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) + ...