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 topic....

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0
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1answer
139 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
117 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 ...
7
votes
5answers
397 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 $$x^4-20x^2+100=4(31+2\sqrt{60}+2\sqrt{90}+2\...
1
vote
2answers
37 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
41 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
80 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 ...
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0answers
80 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 \sqrt[3]...
1
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1answer
24 views

Algebraic Closure terminology doubt

If F and K are fields, what does it mean when we say 'F is algebraically closed in K'?
2
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1answer
73 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
60 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
23 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
vote
0answers
45 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
50 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 $[K:\...
0
votes
2answers
304 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 \mathbb{Q}(\sqrt{2},\...
0
votes
0answers
41 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
103 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 $\mathbb{K}^...
8
votes
1answer
73 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
42 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 ...
1
vote
0answers
30 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 ...
4
votes
0answers
48 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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3answers
83 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 $[K : E] = [K : F][F : E]$. Since $...
2
votes
2answers
65 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 $\Phi:Gal(E/F)\...
0
votes
3answers
339 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 help ...
1
vote
1answer
49 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 $\mathbb{F}...
0
votes
1answer
61 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
32 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
1answer
26 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
40 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
65 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
73 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) (I'...
3
votes
3answers
71 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 \...
2
votes
1answer
48 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
149 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
71 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
83 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
45 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 algebraic ...
4
votes
4answers
165 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
39 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 $F(\...
1
vote
0answers
44 views

Is it possible to define a continuous field with characteristic $\neq 0$?

For example, defining an addition and multiplication on the unit circle in the complex plane such that it forms a field. This would be a sort of continuous analog of the finite fields. Another way I ...
2
votes
2answers
51 views

Show that every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$

A (probably simple) question I encountered but I don't know how to approach: Let $K$ be a field of prime characteristic $p>0$. Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ ...
3
votes
1answer
43 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
0
votes
2answers
54 views

Show $f$ is irreducible.

Let $E$ be an extension field of $F$. Show that if $\alpha \in E$ is algebraic of degree $n$ over $F$ and $f\in F[X]$ is of degree $n$ with $f(\alpha) = 0$, then $f$ is irreducible. For this ...
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vote
1answer
76 views

Every finite abelian extension of Q contains a totally real subfield of index 2?

I can reduce this to the case of cyclotomic field extensions, by embedding the abelian extension into a cyclotomic extension and using the "sliding-up" lemma. I am stuck on how to prove this for the ...
6
votes
1answer
92 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
1
vote
1answer
31 views

the degree of every irreducible polynomial that divides $x^p-x-a$ is the same.

let $F$ be a field with char$(F)=p>0$ where $p$ is a prime.given $a\in F^\times $ ($a\not=0$) denote \begin{equation*}f(x)=x^p-x-a\end{equation*} I'm trying to prove that the degree of every ...
0
votes
2answers
32 views

$\Bbb{Z}_{2}(\alpha)$ as splitting field

i have problems with an exercise: let $\alpha$ be a root of the polynomial $X^{3}+X^{2}+1$ in $\Bbb{Z}_{2}$. Prove that $\Bbb{Z}_{2}(\alpha)$ is the splitting field of this polynomial over $\Bbb{Z}_{...
2
votes
2answers
26 views

If chatacteristic of $K$ is positive, show that every homomorphism from additive group to multiplicative group maps all elements of $K$ to $1$

Suppose $K$ is a field. Denote $(K,+)$ as the abelian group under addition operation and $(K,\times)$ as the abelian group under multiplication opearation. If the characteristic of $K$ is positive, ...
2
votes
2answers
75 views

Maximal ideal in a polynomial ring over a field.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in $\mathcal{K}[...
2
votes
1answer
33 views

Is $\alpha$ a norm in the extension $K(\sqrt[n]{\alpha})$?

I'm having trouble wrapping my head around this. $K$ is a field of characteristic zero containing all $n$th roots of unity, and $\alpha \in K$. Let $L = K(\sqrt[n]{\alpha})$, $\mu$ the minimal ...
6
votes
0answers
93 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...