Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, ...

learn more… | top users | synonyms

3
votes
1answer
143 views

Can extension by an isomorphic field be of degree at least 2?

Suppose $K/F$ is a field extension such that $K\not=F$. Is it legitimate to say that $F$ and $K$ can't be isomorphic since by assumption \begin{equation*}[K:F]\ge 2\end{equation*}and if $K$ and $F$ ...
0
votes
0answers
16 views

Multiplicative inverse of a (non-prime) field

We study field extensions. Two classes widely introduced to newbies of Galois Theory are (isomorphic to, though they may not appear in this form) $\mathbb{Q}^n$ and $\mathbb{F}_p^n$ (for some prime ...
1
vote
1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...
0
votes
0answers
26 views

Choice of Primitive Element in ''Primitive Element Theorem''

Let $F$ be a field of characteristic $0$, and $K$ a finite extension of $F$. Then it is well known (see this) that $K$ can be obtained by attaching an element $\alpha \in K$ to $F$, i.e. ...
2
votes
2answers
64 views

If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible?

Let $F$ be a field of characteristic $p>0$ and $a\in F$. I have an easy question which I'm stuck on. If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$? ...
2
votes
1answer
34 views

Factoring ideals in algebraic number rings using Dedekind's theorem

Let $K \subset L=K(\alpha)$ be a number field extension with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L=\mathcal{O}_K[\alpha]$ respectively. Let $\pi$ be a prime ideal in $O_K$, and let $F = ...
3
votes
0answers
41 views

Ordered fields are real?

This question is motivated by another question posted earlier today. Let $F$ be a field endowed with an embedding $F\hookrightarrow\Bbb R$. Then the ordering of $\Bbb R$ induces an ordering on $F$. ...
0
votes
2answers
25 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
0
votes
1answer
36 views

The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
1
vote
1answer
54 views

Why the extension dimension of $x^3-2$ equal to $6$?

I have seen couple questions related to this one, but after reading the answers I am still confused: Why is the extension dimension of $x^3 - 2$ equal to $6$? In other words, why are the basis ...
2
votes
2answers
41 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
3
votes
1answer
65 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
0
votes
1answer
35 views

Field extension generated by $\alpha$ and separability

In my notes, I have written that the field extension of $k$ generated by an element $\alpha \in K$, where $K$ a larger field, is defined to be $$k(\alpha) = \bigcap_{ \alpha \in E} E$$ where $E ...
5
votes
2answers
35 views

Galois group and degree of splitting field over complex rational functions.

Suppose $F=\mathbb C (t)$ the field of rational functions over $\mathbb C$. let \begin{equation*}f(x)=x^6-t^2\in F[x]\end{equation*}Denote $K$ as the spliting field of $f$ over $F$. I'm trying to ...
0
votes
2answers
41 views

Example check: two algebraically closed fields with one a subset of the other

The problem asks to find two different algebraically closed fields $\mathcal{E}$ and $\mathcal{F}$ with $\mathcal{E} \subseteq \mathcal{F}$. We have not done a whole lot of stuff with algebraically ...
3
votes
0answers
15 views

Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
3
votes
1answer
40 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
5
votes
1answer
24 views

“If $\text{Gal}\left(K/F\right)=\left<\sigma\right>$, $N_{K/F}\left(\alpha\right)=1$ , then $\alpha=\frac{\beta}{\sigma\left(\beta\right)}$.”

For (fintie) Galois extension $K/F$, it is easy to show that $N_{K/F}\left(\frac{\beta}{\sigma\left(\beta\right)}\right)=1$ for all $0\ne\beta\in K$, $\sigma\in\text{Gal}\left(K/F\right)$. I want to ...
4
votes
0answers
43 views

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

I want to find dim. $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$ for positive rational number $a$ and $n^\text{th}$ root of unity $\zeta_{n}$ with assumption ...
0
votes
1answer
24 views

Radical extension and algebraic solution of an irreducible polynomial

Suppose that $k$ is a field with characteristic equal to zero, that $P \in k[X]$ is an irreducible polynomial and that $\alpha$ is a root of $P$ in an algebraic closure $\overline{k}$. Suppose also ...
1
vote
1answer
53 views

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
3
votes
3answers
61 views

Irreducible polynomial over $\mathbb{Q}(\zeta)$

Show that the polynomial $f(x)=x^5-2$ is irreducible over $\mathbb{Q}(\zeta)$, where $\zeta=e^{2\pi i/5}$. I tried show that the roots of polynomial $f(x)=x^5-2$, $$\sqrt[5]{2}, \zeta\sqrt[5]{2}, ...
2
votes
0answers
28 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...
1
vote
1answer
25 views

Isomorphism of quadratic extensions (of a number field)

I think we agree that two (squarefree) quadratic extensions of $\mathbb Q$, say $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ are not isomorphic. Now consider the following tower of fields ...
6
votes
1answer
79 views

Integral basis of $\mathbb{Q}(\theta)$, where $\theta^3-\theta-4=0$

I am working on the text book "algebraic number theory" by Jurgen Neukirch(P15, exercise 6). To prove the integer basis is$ \{1, \theta, \frac{\theta^2+\theta}{2}\}$. After a long and tedious ...
4
votes
2answers
45 views

Degree of minimal polynomial

The minimal polynomial of $a$ over $\mathbb{Q}$ is quadratic. The minimal polynomial of $b$ over $\mathbb{Q}$ is cubic. Is the minimal polynomial of $a+b$ necessarily of degree $6$? If so, what is ...
6
votes
4answers
106 views

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$

I've got that $[\mathbb{Q}(\sqrt{5}+\sqrt[3]{2}):\mathbb{Q}] \in \{1,2,3,6\}$ because it's going to divide $[\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}]=6$. Clearly it is not $1$. I want to show that ...
1
vote
1answer
42 views

Why is is $K(\alpha,\beta)/K(\alpha)$ algebraic if $K(\alpha,\beta)/K(\beta)$ is algebraic? [duplicate]

Let $K$ be a field, and let $\alpha$ be transcendental over $K$ and algebraic over $K(\beta)$. We have a Hasse diagram of field extensions Now, by reduction to absurdity $\beta$ must be ...
2
votes
0answers
28 views

Exercise about cyclic field extension

I am having hard time to solve following exercise. Let $\Omega$ be the algebraic closure of a field $k$. a) Suppose that every finite extension of $k$ is cyclic. Prove that it exists $\sigma \in ...
2
votes
1answer
36 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
1
vote
1answer
48 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
4
votes
0answers
58 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
4
votes
2answers
63 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
2
votes
1answer
35 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
0
votes
1answer
12 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
2
votes
1answer
29 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L ...
1
vote
0answers
32 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
0
votes
0answers
34 views

What is the possible number (supremum) of subfields of $\mathbb{F}$?

Let $\mathbb{F}$ be field. it is a finite dimensional extension over $\mathbb{Q}$. So let $B=\{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb{F}$ over $\mathbb{Q}$. From the finite dimension ...
2
votes
5answers
70 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
4
votes
1answer
50 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
0
votes
1answer
45 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
3
votes
1answer
50 views

Galois group of primitive root of unity/ field extensions

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. I want to determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. Because ...
0
votes
4answers
36 views

Basis of $\mathbb{Q}[\sqrt[3]{2}]$

How do I prove that $1, \sqrt[3]{2}, (\sqrt[3]{2})^2$ is a basis of $\mathbb{Q}[\sqrt[3]{2}] = \{ a + b \sqrt[3]{2} + c (\sqrt[3]{2})^2\: a,b,c \in \mathbb{Q} \}$. It's one of these cases where the ...
4
votes
1answer
49 views

Is the primitve element of $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ always $\alpha_1 + \alpha_2 + \cdots$?

I have dealt with a number of algebraic field extensions $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ and the primitive element was always $\alpha_1 + \alpha_2 + \cdots$. Is this generally true ...
1
vote
0answers
76 views

Extensions of a field?

Prove that there are infinitely many degree 5 extensions of $\mathbb{F}_{121}(x)$. I know that $\mathbb{F}_{121}$ is isomorphic to the splitting field of $x^{121}-x$ over $\mathbb{F}_{11}$, but I'm ...
4
votes
3answers
87 views

What elements may I adjoin to $\mathbb{Q}[\sqrt{3}]$ in order to get to $\mathbb{Q}[\sqrt{7+\sqrt{3}}]$

The field extension $\mathbb{Q}[\sqrt{7+\sqrt{3}}]/\mathbb{Q}$ has degree four and $\sqrt{7+\sqrt{3}}$ is a primitive element. I'm interested in dividing this into two successive field extensions of ...
0
votes
0answers
84 views

Is $\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$ a Galois extension?

Consider the field extension $$\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$$ Is it Galois? I can't quite find the order of the automorphism group.
0
votes
1answer
33 views

Irreducible polynomial of $\sqrt{2}+\sqrt{7}$ on $\Bbb{Q}$.

I would like to find the irreducible polynomial on $\Bbb{Q}$ of $\sqrt{2}+\sqrt{7}$. How can I do that ? First time I see this kind of question, I can find a polynomial $X^2-2$ witch $\sqrt{2}$ is a ...
1
vote
1answer
15 views

Question on the normal closure of a field extension

Hi all I was given the following question in my field theory class which I am stuck on: I am given $ K/F $ a finite extension of fields. I am asked to show the existence of an $ K \subset L $ such ...
0
votes
3answers
42 views

Field extensions for polynomial $T^3+2T +1$ in $\mathbb F_3$

I have the following polynomial over $\mathbb F_3$: $$ f(T) = T^3+2T+1 $$ I would like to find out a field extension in order to add the roots of this polynomial. Edit: by defining $\alpha\not\in ...