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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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
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0answers
46 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
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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 ...
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votes
2answers
36 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
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0answers
27 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
37 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
409 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
36 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
167 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
23 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
22 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
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1answer
45 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 ...
4
votes
1answer
16 views

Let $K \subset L$ be fields and $\tau \in L$ be transcendental over K. Then $\tau$ is algebraic over $K(\alpha)$ for any $\alpha \in K(\tau) - K$

Let $K \subset L$ be fields and $\tau \in L$ be transcendental over K. Show that $\tau$ is algebraic over $K(\alpha)$ for any $\alpha \in K(\tau) - K$ Any help with this question? I'd love a hint ...
1
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0answers
38 views

Does a field of transcendence degree n correspond to a variety?

There's an equivalence of categories between the category of (nonsingular projective) curves over a field $K$ (with dominant morphisms) and finitely generated fields $L/K$ of transcendence degree 1 ...
3
votes
1answer
36 views

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd. If $K/\mathbb{Q}$ is Galois, prove that $K$ is contained in $\mathbb{R}$. Find an extension with ...
0
votes
2answers
40 views

Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.

Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$. Since $\sqrt{2},\sqrt{3}\in \mathbb{Q}(\sqrt{2},\sqrt{3})$, and $\sqrt{2}\sqrt{3}=\sqrt{6}\in ...
0
votes
0answers
28 views

Constructible points from $\mathbb{Q}\times\mathbb{Q}$

I have recently learned the proof for why you cannot "double" the cube, trisect the angle, and "square" the circle. I understand the whole analysis, assuming that a point is constructible if it is ...
3
votes
1answer
74 views

Is there such a norm on any totally disconnected local field?

Let's set this definition of local field: Let $\mathbb{K}$ be a field and a topological space (non-discrete and totally disconnected). Then $\mathbb{K}$ is called a local field if both ...
8
votes
1answer
53 views

Every subring of a field is a domain. Is this reciprocal?

I'm reading my notes on ring theory, and we proved on class that every subring of a field is a domain. Proof: Let $S \subseteq K$ be a subring of $K$, with $K$ a field. Let $x,y \in S$. If $xy=0$, ...
0
votes
1answer
38 views

Determine $[K(\zeta_{16}):K]$ when $K=\mathbb{F}_7, \mathbb{F}_9, \mathbb{F}_{17}$

Let $\zeta_{16}$ be a primitive 16-th root of unity over a field $K$. Determine $[K(\zeta_{16}):K]$ when $K=\mathbb{F}_7, \mathbb{F}_9, \mathbb{F}_{17}$. I know that over $\mathbb{Q}$, the minimal ...
2
votes
0answers
25 views

Irreducible polynomials and poving the ring of intergers is a PID

My question isn't too hard I think I'm just a little stumped on how to tackle the second part. $ Let \ K=\Bbb Q(\alpha)$ where $\alpha$ is a root of $f(x)=x^3+2x+1$ 1) Show that $f(x)$ is ...
2
votes
0answers
39 views

Characterization of tensor products of fields

For which commutative rings $R$ are there field homomorphisms $L \leftarrow K \to L'$ (not assumed to be algebraic or anything) such that $R \cong L \otimes_K L'$? Is there an intrinsic ...
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votes
3answers
75 views

Show $[K : F] = [K : E][E : F]$. [duplicate]

Let $E\subset F\subset K$ be fields. Show that if $K$ is a field extension of finite degree over $F$ and $F$ is a field extension of finite degree over $E$ then $[K : E] = [K : F][F : E]$. ...
2
votes
2answers
33 views

The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Hungerford -Algebra p.271 Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$. Let ...
0
votes
3answers
62 views

Constructing a multiplication table for a finite field

Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$ and let $F=\mathbb{Z}_2(\alpha)$, where $\alpha$ is a root of $f(x)$. Show that $F$ is a field and construct a multiplication table for $F$. Can you please ...
1
vote
1answer
40 views

If $f(x)=x^{m}+1$ is an irreducible polynomial in $\mathbb{F}_{p}[x]$, then prove that $2m$ divides $p^{m}-1$

Here's the full problem: Let $\mathbb{F}_{p}$ denote the finite field of size $p$, where $p$ an integer prime greater than $2$. Suppose that $f(x)=x^{m}+1$ is an irreducible polynomial in ...
0
votes
1answer
48 views

Is the polynomial $x^8+x+1$ irreducible in $\mathbb{F}_2[x]$?

Is the polynomial $f(x)=x^8+x+1$ irreducible inf $\mathbb{F}_2[x]$? I know that if $x^8+x+1$ divides $x^{2^8}-x=x^{256}-x$, then it is irreducible over $\mathbb{F}_2$. I started using the division ...
0
votes
1answer
27 views

A question regarding intersections and products of fields

Does the following hold for $E, L, K$ fields? $$E(L \cap K) = EL \cap EK$$ $$$$ Certainly $E(L \cap K) \subseteq EL \cap EK$ since both $EL$ and $EK$ contain $E$ and $L \cap K$, but I can't see ...
0
votes
0answers
32 views

DVR and its fraction field

Let $k$ be a complete discrete valuation field with algebraically closed residue field. We know that its maximal unramified extension $k^{\mathrm{unr}}$ need not be complete. But can the ring of ...
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votes
1answer
18 views

multiple roots of irreducible polynomial 2

let say we have an irreducible polynomial over field $F$. I need to prove that all roots of f have the same multiplicity. I know that if $\text{Ch}(F)=0$ so this is easy but I don't know what to do ...
2
votes
1answer
29 views

“Any radically closed field contains all roots of unity”

I've seen the statement "any radically closed field contains all roots of unity." Though the term "radically closed field" doesn't seem to be extremely common, I'm fairly confident that it means that ...
3
votes
3answers
37 views

Find a primitive element for the field extension $\mathbb{F}_{125}/\mathbb{F}_5$ - which elements of $\mathbb{F}_{125}$ are not in $\mathbb{F}_5$?

I want to find a primitive element for the field extension $\mathbb{F}_{125}/\mathbb{F}_5$. I constructed $\mathbb{F}_{125}$ as $\mathbb{F}_5[X]/\langle X^3 + X + 1 \rangle$. Since the degree of the ...
0
votes
1answer
38 views

Splitting field of a polynomial of prime degree

Here's the problem stated better: Let $F$ be a field and let $f(x) \in F[x]$ have splitting field $K$. Show that if the degree of $f$ is a prime $p$ and $[K:F]=tp$ for some integer $t$ then (a) ...
2
votes
3answers
46 views

Galois extension of degree $ 2^n $

I'm trying to find a way to prove the following statement: Assume $ \mathbb{Q} \subset E $ is a Galois extension of degree $ 2^n $. Show that there are fields $ \mathbb{Q} = E_0 \subset E_1 \subset ...
1
vote
1answer
25 views

Galois group of $ x^n - a $ over a field containing $ \zeta_n $

I'm having trouble solving an exercise regarding Galois theory. Suppose $ n > 0 $ and $\zeta_n \in F \subset \mathbb{C} $, where $ \zeta_n $ denotes the primitive root of unity of degree $ n $. ...
3
votes
1answer
53 views

Galois group and intermediate fields for splitting field of $ x^3 -7 $

I'm trying to do the following exercise: find the Galois group $ G(E/\mathbb{Q}) $, where $ E $ is the splitting field of $ x^3 - 7 $, all its subgroups and the intermediate subfields $ E^H $ ...
2
votes
1answer
58 views

Why $F=\{p(a) \mid p∈k[x]\}$ is a field?

If $k$ is a subfield of $K$ and $a\in K$. Why the set $F=\{p(a) \mid p∈k[x]\}$ is a field? I think that this is a trivial question but I can't do it by myself.
4
votes
1answer
75 views

Find the field by the its multiplicative group

Suppose we have a group G. Is this a multiplicative (or additive) group of some field? I think that аn arbitrary group is not suitable (e.g. in the case of finite fields multiplicative group should be ...
1
vote
4answers
39 views

Show $α^{ −1}$ is algebraic over $ F $ of degree $n$.

Let $E, F$ be distinct fields such that $E$ is a field extension of $F$. Show that if $\alpha \in E \setminus F$ is algebraic over $F$ of degree $n \in \{2, 3, \cdots\}$, then $α^{ −1}$ is ...
4
votes
4answers
112 views

$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$

Prove that $$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$ I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right ...
3
votes
1answer
27 views

$f(x)$ irreducible in $F[x]$, $\alpha$ a root, show that if some odd degree term of $f(x)$ has nonzero coefficient then $F(\alpha)=F(\alpha^{2})$

Let $F$ be a field, $f(x)$ an irreducible polynomial in $F[x]$ and $\alpha$ a root of $f$ in some extension of $F$. Show that if some odd degree term of $f(x)$ has a nonzero coefficient, then ...