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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3
votes
1answer
48 views

Does there exist $\mathbf{Q} \subset R \subset \mathbf{C}$, $R$ ring & not field

I am looking for an example of a field extension $k \subset F$ and a unital ring $R$ that is not a field such that $$k \subset R \subset F.$$ I know if $F$ is algebraic over $k$, then this is not ...
9
votes
2answers
101 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
3
votes
0answers
31 views

Show that $K(\alpha,\beta)/K$ is simple

Let $L=K(\alpha,\beta)$ be an algebraic field extension, with $\alpha$ separable over K. Show that $L/K$ is simple. My attempt: If we could show that $L/K$ is finite and separable then the claim ...
2
votes
0answers
16 views

Connection between field and the frobenius homomorphism [duplicate]

Let K be a field with char(K)>0. How do I prove that every algebraic extension of K is a separable extension if and only if $\phi:x \rightarrow x^p$ is surjective ?
2
votes
3answers
103 views

Field extensions with(out) a common extension

Let $K$ be a field having two field extensions $L\supseteq K$ and $M\supseteq K$. Does there exist a field $N$ along with embeddings $L\to N$ and $M\to N$, such that the diagram $$ \require{AMScd} ...
0
votes
2answers
38 views

Fixed field of a subgroup of a Galois group

For the Galois group $Gal(\mathbb{Q}(\sqrt2, \sqrt3, \sqrt5)/\mathbb{Q})$, I'm trying to understand how to find the permutations of the roots and how the subgroups of the Galois group are related ...
2
votes
0answers
62 views

$\overline{\mathbb Q}$ and $\mathbb C$ same first order theory

How do you show that $\overline{\mathbb Q}$ (the algebraic closure of $\mathbb Q$) and $\mathbb C$ have the same first order theory over the signature $(0,1,+,\cdot)$?
0
votes
0answers
17 views

Linear algebra over a division ring

As is well known much of the theory of matrices over a field $\mathbb F$ remains correct for matrices over a division ring $D$,(the main exception is the theory of determinants). In which books are ...
0
votes
1answer
20 views

Problem involving a tower of fields with an algebraic and a normal extension

I seem to be stuck on the following problem about field extensions from an old prelim exam in algebra. Let $K$ be an algebraic field extension of a field $F$ and let $L$ be a subfield of $K$ such that ...
1
vote
2answers
56 views

Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$

Let $F$ be a field. Show that $F[x] / (x^2-1) \cong F \times F$ iff $char(F) \neq 2$. Here is my work this far: If $char(F) = 2$, then $x^2-1 = (x-1)^2$, and hence $F[x]/(x^2-1)$ has a non-zero ...
2
votes
1answer
30 views

Field extensions and minimal polynomials

Let $L/K$ be a field extension and $\alpha,\beta \in L$. Let $f\in K[X]$ be the minimal polynomial of $\alpha$ and $g\in K[X]$ be the minimal polynomial of $\beta$ Show the following: $$f \text{ is ...
0
votes
1answer
17 views

What is a requirement for an order of algebraic number field $K$ to be integrally closed domain?

Suppose there is an order $O$, a subring, of an algebraic number field $K$. What is needed (necessary and sufficient condition) for $O$ to be integrally closed domain? Or if we need to impose ...
1
vote
1answer
75 views

Automorphisms of $\mathbb{C}(X)$ and their fixed field

I'm stuck at the very beginning of an exercise I have to do for my algebra class. We're looking at the field of $\mathbb{C}(X)$ and it's automorphisms. Let $a \in \mathbb{C}^*$, $ b \in \mathbb{C} $ ...
0
votes
0answers
19 views

Does $\mathbb{Z}$ in ring of integers $O_K$ in number field $K$ have to be generated by $K$'s multiplicative identity $1_K$?

As title says, does $\mathbb{Z}$ in ring of integers $O_K$ in number field $K$ have to be generated by $K$'s multiplicative identity $1_K$? Or can there be multiplicative identity element ...
-1
votes
0answers
77 views

Transcendental extension and algebraic closure

Let $t$ be a transcendental element over field $K$, and let $F/K(t)$ be finite extension. Assume that $E$ is algebraic closure of $K$ in $F$. How to prove that $E/K$ is finite and $[E:K] \mid ...
0
votes
1answer
33 views

Field Characteristic Is Prime…?

Consider the article: http://mathworld.wolfram.com/FieldCharacteristic.html It is stated that given a field and its multiplicative identity $I_{\times}$ that either: $$ \sum_{i=0}^{k}{I_{\times}} ...
1
vote
3answers
483 views

Extensions of degree two are Galois Extensions.

This question from Allan Clark's "Elements of Abstract Algebra" Show that an extension of degree 2 is Galois except possibly when the characteristic is 2. What is the case when the characteristic is ...
6
votes
0answers
39 views

why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A clasic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only ...
5
votes
1answer
43 views

Can an orderable field always be ordered in a way that extends a given subfield's ordering?

Let $F$ be an orderable field, and let $\:\langle E,\hspace{-0.03 in}\leq \rangle \:$ be an ordered subfield of $F$. Does it follow that $F$ can be made into an ordered field in a way that extends ...
3
votes
3answers
359 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
1
vote
1answer
21 views

What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
0
votes
1answer
38 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
0
votes
1answer
40 views

Prove that factor rings are fields [closed]

Prove that factor rings $\mathbb{Z}_3[x]/(x^3 + x^2 +2)$ and $\mathbb{Z}_3[x]/(x^3 -x +1)$ are fields, and these fields are isomorphic.
-1
votes
2answers
77 views

$\mathbb Q(\sqrt2) \not \cong \mathbb Q(\sqrt[3]{2})$ [closed]

Prove that all fields $\mathbb Q(\sqrt2)$ and $\mathbb Q(\sqrt[3]{2})$ are not isomorphic
0
votes
0answers
19 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
1
vote
1answer
33 views

Intermediate fields of $X^p - 2 $

I've been working on an exercise I have to do for my algebra course. Exercise: Let p be prime and $L$ the splitting field of $ f = X^p - 2$ over $\mathbb{Q}$. a) Show that $ Gal(L/\mathbb{Q})$ is ...
0
votes
2answers
56 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
0
votes
2answers
76 views

What is the splitting field of $t^4+2$?

I currently beginning the study if Galois theory but my understanding of the construction of splitting fields could be better. So I must ask if I could see the steps in constructing the splitting ...
1
vote
2answers
80 views

Simple extension fields

If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as ...
5
votes
1answer
77 views

Prove that $k(\alpha+\beta)=k(\alpha,\beta)$

I am trying to solve the following problem: Let $k$ be a finite field and let $k(\alpha,\beta)/k$ be finite. If $k(\alpha)\cap k(\beta)=k$, prove that $k(\alpha,\beta)=k(\alpha+\beta)$. What I ...
1
vote
1answer
37 views

Specific question on imaginary quadratic field [closed]

How to solve the following question?! Let $K$ be an imaginary, quadratic field and let $L/K$ be a Galois extension. If $\tau$ is complex conjugation, show that: (a) $L/\Bbb Q$ is Galois iff ...
2
votes
3answers
195 views

Showing Galois Group is Abelian

I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to ...
11
votes
2answers
184 views

Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$.

Is $\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$? Say $L=\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)$ and $K=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$. It is easy to show that ...
4
votes
2answers
51 views

When is a number square in Galois field p^n if it's not square mod p?

Here is the problem, that I'm stuck on. There is no square root of $a$ in $\mathbb{Z}_p$. Is there square root of $a$ in $GF(p^n)$? Well, it's certainly true that $$x^{p^n}=x$$ and $$x^{p^n-1}=1$$ ...
7
votes
2answers
108 views

Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
8
votes
2answers
167 views

How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$?

Let $p>2$ be an odd prime. Let $\mathbb F_{p^n}$ be the field with $p^n$ elements. How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$? My work: Char $F=p$. ...
4
votes
1answer
51 views

Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
3
votes
1answer
67 views

Find all the fields between $\mathbb{Q}$ and the splitting field of $x^4 + 81$

Let $f(x)=x^4+81 \in \mathbb{Q}[x]$. Find the splitting field $E$ of $f(x)$ and the extension degree $[E:\mathbb{Q}]$. Find all the fields $L$ with $\mathbb{Q} \leq L \leq E$. Are the roots of ...
1
vote
2answers
297 views

Finding the minimal polynomial in this field extension of $\mathbb Q$?

I have a field extension $K = \mathbb Q[x]/(x^2 - 5)$ of $\mathbb Q$, and an element $a = \bar x \in K$. I need to find the minimal polynomial of $a$ over $\mathbb Q$. I have worked out that ...
14
votes
2answers
1k views

How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?

This is Exercise 18.14 from Algebra, Isaacs. $p_{1}\ ,\ p_{2}\ ,\ ... p_{n}$ are different prime numbers. How to show that $$\mathbb{Q}[\sqrt{p_{1}}, \sqrt{p_{2}}, \ldots, \sqrt{p_{n}} ] = ...
2
votes
0answers
73 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...
0
votes
0answers
9 views

Explanation of why a differential field extension must be trancendental

I'm attempting to follow the steps of this proof that $e^{x^2}$ has no antiderivative and there is one step that I'm not quite understanding. They state: If K is a differential field then $K_C=\{r ...
0
votes
1answer
22 views

Find extension degree

Let $\zeta =e^{\pi i/12}$. Find the extension degree of $\mathbb{Q}\leq \mathbb{Q}(\zeta)$ Show that $\mathbb{Q}(\zeta)=\mathbb{Q}(\sqrt{2} , \sqrt{3} , i)$ $\zeta$ is a root of $x^{24}-1$ ...
0
votes
1answer
42 views

Show that it is an element of $L$

Let $L$ the subfield of the complex with $\mathbb{Q}\leq L$ a normal extension. If $a=\sqrt{5}-\sqrt[3]{2}\in L$, show that $\omega=e^{2\pi i/3}\in L$. I have done the following: ...
2
votes
0answers
38 views

If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
1
vote
0answers
32 views

The equation has a solution iff $(k,q-1)=1$

Let $p^n$, where $p$ is a prime, and $k \in \mathbb{N}$. Let $a$ the generator of the cyclic group $\mathbb{F}_p^{\star}$, where $\mathbb{F}_q$ the finite field with $q=p^n$ elements. Show that the ...
0
votes
1answer
49 views

$X$ is algebraic over $E$

Let $X$ be transcendental over a field $F$, and let $E$ be a subfield of $F(X)$ properly containing $F$. Prove that $X$ is algebraic over $E$. Could we maybe use also the following?? Let $f(x) \in ...
0
votes
1answer
27 views

The polynomial splits into distinct factors in $F[x]$

Let $F$ be a field of characteristic $p$. Show that if $x^p-x-a$ is reducible in $F[x]$, then it splits into distinct factors in $F[x]$. I have done the following: We want to show that ...
1
vote
2answers
28 views

$G$ is isomorphic to $S_3$

Show that the Galois group of the splitting field $F$ of $X^3-7$ over $\mathbb{Q}$ is isomorphic to $S_3$. I have found that the the Galois group is the following: $$G=\{\tau_{ij}, i=1,2,3, ...
1
vote
1answer
25 views

Finite fields and order [duplicate]

If $F$ is a finite field then is it necessary that $|F|=p^n$ for some prime $p$ and positive integer $n$? I know that given prime $p$ and positive integer $n$, there is a field such that $|F|=p^n$? ...