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, ...

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2
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1answer
31 views

Primitive element and Galois group Computation

Let $L=\mathbb{Q}(\sqrt{2},\sqrt{3},y)$ where $y^2=(9-5\sqrt{3})(2-\sqrt{2})$. Show that $y$ is a primitive element of the extension $L/\mathbb{Q}$. Determine its minimal polynomial and a splitting ...
0
votes
1answer
17 views

Construct the Normal Closure for this extension

I am trying to find the normal closure for the following extension $\Bbb Q\subset\Bbb Q(t)$, where "$t$" is a zero of $x^3-3x^2+3$ and $\Bbb Q$ are the rational numbers. I know the normal closure ...
0
votes
3answers
28 views

Finite fields and isomorphism

For each prime number p, let $F_p$ denote the field of integers modulo p. Now let K be any finite field. a) Prove that K contains a subfield isomorphic to $F_p$ for some prime number p b) Prove that ...
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0answers
44 views

If $a+b$ and $ab$ are algebraic then $a,b$ are algebraic

I need to show that if the sum and the product of two complex numbers is algebraic, then each of then is algebraic. We have that the extensions Q(a+b)/Q and Q(ab)/Q are finitie, so the extension ...
-4
votes
3answers
26 views

If α is algebraic over K then all the elements of K(α) are algebraic over K [on hold]

Let $L$ : $K$ be a field extension, $\alpha \in L$ algebraic over $K$. Show that every element of $K(\alpha)$ is algebraic over $K$, where $K(\alpha)$ is the smallest subfield that contains $\alpha$ ...
2
votes
1answer
27 views

$a$ and $1+a^{-1}$ have same degree over $F$ if $a$ is algebraic over $F$

Suppose that $a$ is algebraic over a field $F$. Show that $a$ and $1+a^{-1}$ have the same degree over $F$. Not really sure where to start for this one. I know that I have to show that ...
0
votes
2answers
34 views

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$.

Let $a$ be a complex zero of $x^2+x+1$ over $\mathbb{Q}$. Prove that $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(a)$. Let $f(x)=x^2+x+1$. Then $f(a)=a^2+a+1=0$. To show equality of the two fields, we need ...
1
vote
2answers
42 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
0
votes
1answer
17 views

Q basis for splitting field

I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it ...
0
votes
1answer
18 views

prove $[E(a):E] \le [F(a):F]$

Let $F<E<K$ be field extensions, such that $a \in K$, and $[K:F]<\infty$, Is it true that $[E(a):E] \le [F(a):F]$? How can I show this?
1
vote
1answer
20 views

Question about field extension notation

Hello all I was given the following question about which I understand everything except possibly the notation. I am given a sub-field $ F \subseteq R $ and I am asked to prove the degree of the "field ...
1
vote
1answer
20 views

Prove that there are no separable extensions of $k$ of degree $n$

Let $k$ be a field and let $n \gt 0$ be an integer. Assume that there are no irreducible polynomials of degree $n$ in $k[x]$ . Prove that there are no separable extensions of $k$ of degree $n$ I ...
0
votes
1answer
10 views

About relation between degree of extension and normality

Hi i know that if $F<E$ is an field extension and $[E:F]=2 $ then it is normal extension. How can i show that if $[E:F]=k$ for any $k>2 $ doesnt imply E is normal extension. Basically i need ...
1
vote
2answers
60 views

Field $K (x)$ of rational functions with coefficients from $K$, if $f\in K(x)$, then $f^2 \neq x^2-1$

I'm in the process of studying for an exam and I came across the following question: Prove that if $K$ is a field and $K (x)$ is the field of rational functions with coefficients from $K$, if ...
-5
votes
0answers
27 views

The zeroes in $\mathbb{C}$ of $x^{28}-1 ∈\mathbb Q[x]$ form cyclic group under multiplication [on hold]

The zeroes in $\mathbb{C}$ of $x^{28}-1 ∈\mathbb Q[x]$ form cyclic group under multiplication, explain this.
1
vote
1answer
30 views

Showing two field extensions are equal

Let a,b $\in$ $\mathbb Q$ with b nonzero. Show that $\mathbb Q$($\sqrt a$)=$\mathbb Q$($\sqrt b$) if and only if $\exists$ c $\in$ $\mathbb Q$ such that a=b$c^2$. I am confused on how it is possible ...
1
vote
1answer
49 views

A problem of field in abstract algebra

If $V$ is a finite-dimensional vector space over the field $K$, and if $F$ is a subfield of $K$ such that $[K:F]$ is finite, show that $V$ is a finite-dimensional vector space over $F$ and that ...
2
votes
1answer
18 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
1
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0answers
24 views

When is the normality of field extensions transitive?

I want to answer the following question: Given field extensions $K \subset M \subset L $, when is the case that $M:K$ is normal and $L:M$ is normal implies $L:K$ is normal? In my algebra class we have ...
2
votes
1answer
51 views

When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?

Let $x\in\mathbb{C}$, and consider the field $\mathbb{Q}(x)$. If this field is a finite extension of $\mathbb{Q}$, then $x$ is algebraic over $\mathbb{Q}$, so satisfies some polynomial $P$. Any field ...
-1
votes
3answers
71 views

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ [on hold]

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ . and write down the conjugates of the number over $\mathbb{Q}$
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votes
1answer
14 views

Field Extension and basis of polynomial set [on hold]

Let $L/K$ be a field extension, $\alpha \in L$ algebraic over $K$. Assume $K \subseteq L$. Then with $n = deg m_{\alpha}(k) $, Show that ${1,\alpha,\alpha^2,...,\alpha^(n-1)}$ is a basis for ...
1
vote
1answer
52 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
2
votes
1answer
23 views

Finite field extensions - $K(\alpha)$

So I am currently studying Algebraic Number theory and a theorem in the Book states the following: Let $L/K$ be a field extension. Then $\alpha \in L$ is algebraic over $K$ if and only if there is ...
0
votes
1answer
32 views

Finite field extension over the rationals need only one generator?

Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the ...
0
votes
1answer
77 views

Extend ${\bigl(1+\frac1x\bigr)}^{{x}}$ to $\overline{\mathbb R}$

We can extend these functions to $\overline{\mathbb R}$ by taking limits says here. \begin{align} \mathrm e^{-\infty} &= 0 \\ \mathrm e^{+\infty} &= \infty \\ \ln{\left|0\right|} &= ...
0
votes
1answer
40 views

Degree of minimal polynomial over $\mathbb{Z}_7$

While working through my book I've run into a question where I'm not too sure what is being asked of me/how to start thinking about it. It states: Suppose $E$ is an extension field of ...
0
votes
3answers
27 views

Degree of extension fields

I've been working my way through some problems to get this idea solidified but now I've run into something odd. How do I find the degree of $\mathbb{Q}(\sqrt{1+ \sqrt{2}})$ over $\mathbb{Q}$? Would it ...
1
vote
2answers
69 views

Finding degree of minimal polynomial of $a$ over $F$

So while working through Abstract Algebra by Dan Saracino I've run into some confusion. I'm on a question that states the following: Suppose $E$ is an extension field of $\mathbb{Z_5}$ and $E$ has ...
-2
votes
0answers
36 views

Algebraic extensions help?

$K$ is an extension field of $F$. If $[K : F]$ is finite and $u$ is algebraic over $K$, prove that $[F(u) : F]$ divides $[K(u) : F]$.
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0answers
26 views

The discriminant of (1,x,x^2) in a cubic field.

Let $K$ be a cubic field such that $K=\mathbb Q[x]$ with $x^3=2$. The discriminant of $\{1,x,x^2\}$ is supposed to be $\begin{vmatrix} 3 & 0 & 0 \\ 0 & 0 & 6 \\ 0 & 6 & ...
1
vote
2answers
58 views

$i \notin \mathbb{Q}[\sqrt[4]{2}]$ without using topological properties of $\mathbb{R}$

I can think of two related ways to prove that $i \notin K = Q[\sqrt[4]{2}]$: $K$ is a subset of the real numbers and $i$ is not a real number. $K$ is orderable and no ordered field can contain ...
1
vote
1answer
65 views

is that true $\mathbb{R}(a, b) = \mathbb{R}(a)( b) $? [closed]

is that true $\mathbb{R}(a, b) = \mathbb{R}(a)( b) $ ?
4
votes
1answer
109 views

Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite ...
1
vote
2answers
65 views

Rigorously prove $\mathbb{Q}(\sqrt2,\sqrt3) = \mathbb{Q}(\sqrt2) (\sqrt3)$

As stated, I want to argue that the identity holds i.e. the smallest field containing $\sqrt2$, $\sqrt3$ and $\mathbb{Q}$ is indeed the smallest field containing $\sqrt3$ and $\mathbb{Q}(\sqrt2)$.
0
votes
1answer
37 views

Why are ring extensions only discussed in the context of $\mathbb{C}$?

I'm watching the great (imo) set of lectures on abstract algebra from the harvard extension school that's available on youtube. Now this lecture is about extending a ring. The lecturer talk about ...
0
votes
1answer
28 views

Extensions of fields and dimension

The following is a homework question: Let M: N be a field extension, with a ∈ M algebraic over N. Show every element of N(a) is algebraic over N. Can anyone give me a strategy to approach this?
1
vote
1answer
39 views

If $I$ proper ideal of $R$, $S$ ring extension of $R$, and $u$ a unit in $S$, then $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$

Let $R ⊆ S$ be an extension of rings, and let $u$ be a unit in $S$. Let $I$ be an ideal of $R$ with $I \ne R$. Show that $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$. Here is what I try: I have ...
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0answers
24 views

prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$

$\mathbb{Q}<\mathbb{F}<\mathbb{C}$ - field extensions, such that $[\mathbb{F}:\mathbb{Q}]=m \in \mathbb{N}$ p is a prime number, $\alpha=p^{\frac{1}{n}}$ gcd(m,n)=1 prove n divides ...
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0answers
27 views

Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
0
votes
1answer
29 views

an “explicit” extension field that contains a root of an irreducible polynomial

There is a famous theorem saying that Let $\Bbb F$ be a field and $f(x)$ an irreducible polynomial in $\Bbb F[X]$. Then there exists a field extension $\Bbb L$ of $\Bbb F$ such that $f(x)$ has a ...
2
votes
1answer
47 views

Determine whether the extension is Galois [duplicate]

I am trying to prove that $K=\mathbb{Q}(2^{1/3}, i\sin{2\pi/3})$ is Galois extension over $\mathbb{Q}$. It is easy to see that $K=\mathbb{Q}(2^{1/3},i\sqrt{3})$. I know it is Galois since $K$ is a ...
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0answers
11 views

Question on degree of field extension

Hello all I have the following question in field theory HW assignment and would really appreciate any help. We are given a field F and two extensions of it such that the following holds: ...
0
votes
3answers
43 views

If field $K/F$ is generated by the $\alpha_1,…,\alpha_n$, then an $\sigma\in $ Aut$(K/F)$ of $K$ uniquely determined

Does this proof seem correct? I'm having second doubts concerning the bolded material. Show that if the field $K$ is generated over $F$ by the elements $\alpha_1,...,\alpha_n$, then an automorphism ...
0
votes
1answer
21 views

Automorphism in splitting field

Suppose $F\subseteq L$ is any field extension, $f(x) \in F[x]$ and $\beta_1,\beta_2,....\beta_r\in L$ are distinct roots of $f(x)$. Prove a)If $\sigma$ is an automorphism of L that leaves F fixed ...
2
votes
1answer
52 views

Is every isomorphism of an algebraically closed field onto one of its subfields an automorphism?

I've just been reading about the Isomorphism Extension Theorem, and I think I can make the following argument: Let $F$ be an algebraically closed field, and let $\sigma$ be an isomorphism of $F$ onto ...
1
vote
1answer
72 views

Dimension of union of fields over intersection of fields.

What is the dimension of the union of fields $L_1$ and $L_2$ over $L_1 \cap L_2$ ($L_1$ and $L_2$ have dimensions $n_1$ and $n_2$ over $L_1 \cap L_2$)as a vector space? I think it should be $n_1n_2$ ...
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vote
2answers
32 views

A question on on field extensions and minimal polynomial

Hello I am a novice in fields and was asked this question in an assignment: I need to find the minimal polynomial of the expression $\sqrt[3]{7-\sqrt{2}}$ over the rationals Q Here is where I am ...
0
votes
0answers
32 views

If L/K is algebraic field extension, is L(B)/K(B) also algebraic?

Seems a naive question, but I get stuck there.. Say $M/L/K$ is a tower of fields, $L/K$ is algebraic, $B\subset M$. ($B$ is not necessarily algebraic independent over $K$ or $L$.) Is $L(B)/K(B)$ ...
1
vote
2answers
28 views

Minimal polynomial problem

Show that $\mathbb Q(\sqrt 2 + i)=\mathbb Q(\sqrt 2, i)$ and find minimal polynomial. My question: Assume that they are equal, then the minimal polynomial of both sides must be the same. To prove and ...