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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2
votes
1answer
32 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
33 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 ...
1
vote
0answers
38 views

What does this theorem mean, exactly?

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $F_p[X]$ of degree $d$ where $d$ runs through all the divisors of $n$. I don't even get the ...
0
votes
0answers
8 views

Question on separable degree

I asked a similar question before but I didn't get a satifying answer, so I'm posting it again. Let me first define terms: Def1 Let $E/F$ be an algebraic field extension and $\bar F$ be an ...
7
votes
1answer
78 views

If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.

Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + ...
0
votes
0answers
36 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 ...
-1
votes
1answer
47 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
3
votes
3answers
100 views

$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in ...
1
vote
1answer
29 views

Finding proper subfields

Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly. Two of ...
0
votes
1answer
33 views

Relation between $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$

I have the following exercise in my homework: Are $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$ isomorphic, where $w = \textrm{cis}((2\pi)/3)$? Prove your answer. I think they are, but I'm ...
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 ...
0
votes
1answer
32 views

Is the statement of the theorem correct?

I have been asked to prove this:: $f,g$ are polynomials over a field $F$ .Prove that if $f,g$ are relatively prime then $f,g$ have no common roots in any extension of $F$. But I wonder why is ...
4
votes
2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
4
votes
1answer
51 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
50 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 $X^3-2$ ...
1
vote
1answer
45 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...
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 ...
8
votes
1answer
72 views

On the existence of field morphisms

Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
0
votes
1answer
33 views

Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
0
votes
2answers
26 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
2
votes
3answers
111 views

How do I prove this field homomorphism is an isomorphism?

The question is as follows. Let $F$ be a finite field with unit $1$ not equal to zero. Let the function $f: F \to F$ be given by $f(x) = x^3$, where the $\operatorname{char}(F) = 3$. Prove it is a ...
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 ...
2
votes
2answers
40 views

Injectivity and norm function on finite fields [closed]

Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by $$f(x)=\alpha x^q+\alpha^q x$$ for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 ...
2
votes
1answer
14 views

A basis of a field extension $K \subset L$ spans $L(C)$ over $K(C)$ for any subset $C$ of a field $M\supset L$

Let $K \subset L \subset M$ be fields; $\{\beta_1, ..., \beta_k\}$ a basis for $L$ over $K$ and $C$ a subset of $M$. Then $\{\beta_1, ..., \beta_k\}$ generates $L(C)$ over $K(C)$ (where $K(C)$ is the ...
6
votes
3answers
90 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
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 ...
1
vote
0answers
19 views

Roots of a polynomial over a finite field

Let $f(x)=a_0x+a_1x^q+... a_{k-1}x^{q^{k-1}}$ be a nonzero polynomial for a prime $q$. It is easy to observe that $$f:F_{q^n}\to F_{q^n}$$ a linear function. I want to show that $f$ has at most ...
4
votes
1answer
77 views

Finite fields and their subfields

Let $\mathtt{F}$ and $\mathtt{F'}$ be two finite fields of order $q$ and $q'$ respectively. Then: $\mathtt{F'}$ contains a subfield isomorphic to $\mathtt{F}$ if and only if $q\le q'$ ...
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
0answers
48 views

Theorem with splitting fields

I am trying to understand the following: Theorem I. If the polynomial $p(x)$ is irreducible in $F[x]$ and if $a$ is a root of $p(x),$ then $F(a) \cong F'(b)$ where $b$ is a root of $p'(t) \in ...
3
votes
1answer
27 views

What do the statement mean by "leaves every element of $F$ fixed?

If $p(x) \in F[x]$ and $a,b$ are both roots of an irreducible polynomial $p(x),$ then $F(a) \cong F(b)$ by an isomorphism which takes $a$ onto $b$ and leaves every element of $F$ fixed. Simple ...
0
votes
2answers
37 views

Determine the degree of the splitting field of the polynomial $x^4-2$

Determine the degree of the splitting field of the polynomial $f(x) = x^4-2$ over $\mathbb{Q}$ My attempt: The roots of $f(x)$ are $2^{1/4},-2^{1/4}, 2^{1/4} e^{ \pi i/2} $ and $-2^{1/4} e^{ \pi ...
1
vote
0answers
31 views

Being Galois is not transitive

Let $F\subseteq B\subseteq E$ be fields. I want to prove that if $E/B$ and $B/F$ are Galois, then $E/F$ need not be Galois. Hint: ...
0
votes
1answer
38 views

$f(x)\in K[x]$ implies $\deg(f)\mid [E:F]$

Let $F$ be a field, $f(x)\in F[X]$ irreducible, $n$ the degree of $f(x)$, and $E/F$ the splitting field of $f(x)$. I want to prove that $n\mid [E:F]$. I try this by induction. $n=1$ is trivial. ...
0
votes
1answer
36 views

Describe all $p^{n}$ (in terms of congruence conditions of $p$ and $n$) for which $x^{2}+1$ irreducible over $\mathbb{F}_{p^{n}}$.

So I've said $x^{2}+1$ is reducible over $\mathbb{F}_{p^{n}} \iff \mathbb{F}_{p^{n}}$ contains a root $\alpha$. Hence if $\alpha$ is such a root then $\alpha^2 = -1$ so that $\alpha^4 =1$ and hence ...
2
votes
0answers
36 views

Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse ...
1
vote
2answers
28 views

field of characteristic $p$ be algebraically closed?

Can a field of characteristic $p$ be algebraically closed? I know finite fields cannot be algebraically closed, but there are also infinite fields of characteristic $p$, so can they be algebraically ...
33
votes
2answers
415 views

Shortest irreducible polynomials over $\Bbb F_p$ of degree $n$

For any prime $p$, one can realize any finite field $\Bbb F_{p^n}$ as the quotient of the ring $\Bbb F_p[X]$ by the maximal ideal generated by an irreducible polynomial $f$ of degree $n$. By dividing ...
1
vote
1answer
19 views

$[K:K\cap \mathbb{R}]$ for $K$ a Galois extension of $\mathbb{Q}$

Suppose $K$ is a Galois extension of $\mathbb{Q}$, the field of rational numbers. How do I prove that $[K:K\cap \mathbb{R}] \leq 2$, where $\mathbb{R}$ denotes the field of real numbers? I could ...
0
votes
1answer
41 views

Perfect field of characteristic $p$

I want to prove that a field $F$ of characteristic $p$, is perfect if and only if every element in $F$ has a $p$th root in $F$. We say that $F$ is perfect if every polynomial $f(x)\in F[x]$ is ...
2
votes
3answers
101 views

Algebraic numbers are a field

I want to prove that algebraic numbers are a field using extensions field theory. This seems to be very easy, so I feel strange for not understanding this. The exercise says: let $E/F$ be an ...
5
votes
5answers
179 views

Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$

To find the above minimal polynomial, let $$x=\sqrt{2}+\sqrt{3}+\sqrt{5}$$ $$x^2=10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}$$ Subtracting 10 and squaring gives ...
1
vote
2answers
26 views

Constructibility of roots of a polynomial

I`m trying to decide if the roots of the polynomial $f(x) = x^4+x^3-2x^2 +x +1$ is constructible. My first thought was to show that the polynomial f is irreducible in $\mathbb{Q}$ then for any root ...
3
votes
1answer
36 views

Prove if $L = K(α_1, . . . , α_r)$ and each $\alpha_i$ is separable over $K$, then $L/K$ is separable

Let $L/K$ be a finite extension, $[L:K] = n$. Prove the following are equivalent: $L/K$ is separable $L=K(\alpha_1,...,\alpha_n)$ and every $\alpha_i$ is separable over $K$. I ...
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
0answers
24 views

Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$

I am trying to find the Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$. The roots of this polynomial are $\pm \sqrt{2}$, $\zeta_3^k \sqrt[3]{2}$, and $\zeta_3^j ...
1
vote
1answer
23 views

Algebraic Closure terminology doubt

If F and K are fields, what does it mean when we say 'F is algebraically closed in K'?
1
vote
1answer
47 views

If $p(x)\in F[x]$ is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots.

I want to prove that if $p(x)\in F[x]$, where $F$ is a field, is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots. I found this argument in a book, but I don't ...
0
votes
3answers
57 views

Problem in understanding a theorem

I want to understand this : Let $f$ be a nonconstant polynomial over the field $F$. Then there is an extension $E/F$ and an element $\alpha \in E$ such that $f(\alpha ) = 0.$ Proof. I have no ...