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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2answers
21 views

Abstractly constructing splitting fields

I have a series of exercises where I have to determine the degree of various splitting fields. I am freely using the following observation, which I feel is intuitively true, but I am asking here to ...
1
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1answer
26 views

An algebraic element $a$ in a field extension $K/F$ satisfies $a^{q^m}=a$

Let $F$ be a field with order $q$ and characteristic $p$. Show that if $a$ is an algebraic element over $F$ in the extension $K$, then $a^{q^m}=a$ for some $m$. I have shown that the order of the ...
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0answers
23 views

How can one show algebraically that an angle is constructible?

For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. ...
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0answers
22 views

$f(x) = x^2 + bx + a$ irreducible over $\Bbb F_p$ (finite field of $p$ prime elements) iff $(b^2 - 4a)^{\frac{p-1}{2}} = -1$ in $\Bbb F_p$

My attempt started as follows. I know that for $f$ to be irreducible, $D = b^2 - 4a$ is not a square in $\Bbb F_p$ (ie $(\frac{D}{p}) = -1$). I also know that $D^{p-1} = 1$, so I see $\sqrt{(D^{p-1})} ...
1
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0answers
34 views

Structure of Galois group

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$ and $K|_F$ be the splitting field of $f(x)$. I want to prove the statement that if $G = \text{Gal}(K|_F)$ is cyclic then $[K:F] ...
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1answer
33 views

$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
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1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
2
votes
2answers
46 views

Proving $f(x)$ is not a square in $k[x]$

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
1
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1answer
39 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
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5answers
85 views

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

Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ and find all $w\in \mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ such that $\mathbb{Q}(w)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. It ...
4
votes
1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
0
votes
1answer
23 views

Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
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2answers
58 views

$\operatorname{char}R=0 \implies\mathbb{Q} \hookrightarrow R$

Let $R$ be any field, then: $$\operatorname{char}R=0 \implies \mathbb{Q} \hookrightarrow R$$ Proof: We know that $\mathbb{Q} = Q(\mathbb{Z})=\{[(x,y)]\subseteq\mathbb{Z}\times \mathbb{Z^*}:(x,y) ...
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2answers
48 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
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2answers
35 views

Splitting field of $x^5-3x^3+x^2-3$

I am trying to solve the following problem, Find the degree of the splitting field of the polynomial $p(x)=x^5-3x^3+x^2-3$ over $\mathbb{Q}.$ My approach for solution: Clearly -1 is a root of the ...
2
votes
2answers
42 views

If Q(a,b) is a field extension, can we always choose an equivalent extension Q(c) such that c=a+b?

If we have two complex numbers $a,b$ that are algebraic over $\mathbb {Q} $, we can make an extension $\mathbb {Q}(a,b)$ that is equal to an extension $\mathbb {Q}(c)$ for some $c\in \mathbb {C} $. ...
3
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0answers
31 views

Infinite extensions of “finite degree under $\mathbb{Q}$” [duplicate]

Consider an algebraic extension $K$ of $\mathbb{Q}$. The degree $[K:\mathbb{Q}]$ of $K$ is defined as the dimension of the extension considered as a vector space. Now, let $\overline{\mathbb{Q}}$ be ...
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0answers
23 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
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0answers
17 views

Frobenius Map and Subfields of $\bar{\mathbb{F}}(x,y)$

Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) ...
1
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1answer
38 views

Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
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1answer
41 views

$\mathbb{Z}_p \hookrightarrow R$ is it necessary $R$ commutative?

Let $R$ be an integral domain with $\operatorname{Char}(R)=p$, with $p$ prime. Then: $$\mathbb{Z}_p \hookrightarrow R$$ The proof is not difficult. My questions are: 1) Is it necessary to have an ...
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1answer
43 views

$X^4-10X+1$ reducible in $\mathbb{F}_p[X]$ for all prime $p$ [duplicate]

Show that the polynomial $X^4-10X+1$ is irreducible in $\mathbb{Z}[X]$ but reducible in $\mathbb{F}_p[X]$ for all prime $p$. I could show the irreducibility in $\mathbb{Z}[X]$ but not sure how to ...
2
votes
0answers
28 views

char$(K)=0$ then $K(x^{2}) \cap K(x^{2}-x)=K$ [duplicate]

Let $K$ be a field of characteristic zero and $K(x)$ the field of rational functions with coefficients in $K$. Let $K(u)$ denote the subfield of $K(x)$ generated by $u \in K(x)$ over $K$. My ...
3
votes
2answers
59 views

Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
6
votes
1answer
158 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...
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2answers
25 views

In a field of characteristic 0, for any integer $m$ and an element $x$, does there exist another element $y$ that $ym=x$?

As the title. Or rather, for any integer $m$ which is not the characteristic, does such an 'integer division' exist?
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0answers
39 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
0
votes
0answers
14 views

A normal closure of an arbitrary field extension

Let $L/K$ be an arbitrary algebraic field extension. How is a normal closure of $L$ (the smallest normal extension of $K$ containing $L$) constructed? If $L/K$ is finite, then writing ...
0
votes
1answer
21 views

Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Let $F$ be an extension field over $K$ such that $F(x)$ is a finite extension over $K(x)$ ; then is it true that $[F:K]$ is also finite ? ( I know about the converse , that if $F/K$ is a finite ...
0
votes
1answer
65 views

If $x^p−x−c$ is irreducible in $F[x]$ then it has no root in the field.

The complete problem appears in Hungerford's Algebra. Let $c\in F$, where $F$ is a field of characteristic $p$ ($p$ prime). Then $x^p−x−c$ is irreducible in $F[x]$ if and only if $x^p−x−c$ has no ...
0
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1answer
16 views

Proving an element belongs to field extension

I am unsure of questions asking to prove that an element belongs to a field extension. Here is an example: Prove that $\sqrt2 \in \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$ $\sqrt2 \notin ...
2
votes
2answers
39 views

Normal closure of $\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$

The following is a question from an undergrad course in Galois theory: Find a normal closure $L$ of $K=\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$ I know that normal extensions are ...
3
votes
1answer
31 views

If $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ and $[K : \mathbb{Q}]=p$ then $K= \mathbb{Q}(\sqrt[p]{2})$

Let $p,q$ be distinct prime numbers and assume that $p<q$. Prove that if a subfield $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ satisfies $[K : \mathbb{Q}]=p$, then $K= ...
0
votes
0answers
17 views

Normal transcendental extension

According to Wikipedia, normal extension are assumed to be algebraic. But one of the definitions $K/k$ is normal if any $k$-embedding $\sigma : K \rightarrow \Omega$ of $K$ into a fixed algebraic ...
2
votes
1answer
40 views

Example of finite field extension where root not separable

My class notes has as theorem (without proof): "Let $K/F$ be finite field extension, with $K=F(\alpha_1,\ldots,\alpha_n)$ and $\alpha_k$ is separable for all $k$. Then $K/F$ is separable". My ...
0
votes
1answer
29 views

If every polynomial in $F[x]$ splits then there exists no nontrivial algebraic extension

Im trying to prove the statement of the title: If every polynomial in $F[x]$ splits then $F$ has no nontrivial algebraic extension I was thinking about arguing as follows: if there existed an ...
0
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0answers
26 views

Complex Norms when D = 1 mod 4

Let $D ∈ \mathbb Z$ and let $\alpha ∈ \mathbb C$ be such that $\alpha^2 = D$. Let $\beta = \frac{1+\alpha}{2}$ and $\overline{\beta} = \frac{1-\alpha}{2}$ if $D = 1$ mod $4$ and $\beta = \alpha$, ...
1
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1answer
50 views

There is no field with exactly 6 elements

I saw the related posts, and I tried a different proof. Please have a look. Let $D$ be any field with $|D|=6$. $|D|=6<\infty \Longrightarrow Char(D)\neq 0\Longrightarrow Char(D)=prime\ number$ ...
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0answers
22 views

Find the degree of a tower of field extensions

Let $E = F(\alpha, \beta)$ be an extension of the field $F$. We're given that the minimal polynomial of $\alpha$ in $F[x]$ is of degree $d_1$, and the minimal polynomial of $\beta$ in $F[x]$ is of ...
0
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0answers
15 views

$ \forall a\in U(R) : ord(a)=Char(R) $

Theorem: Let $(R,+,\cdot)$ be a ring with unity $1_R$. Then $$ \forall a\in U(R) : ord(a)=Char(R) $$ Proof: If $ord(a)=n$, $ord(1_R)=m=Char(R)$ then $n1_R=n(a \cdot a^{-1})=(na) \cdot a^{-1}=0_R ...
0
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2answers
21 views

Suppose $\gamma$ is the root of some irred. polynomial in F[x], why is [F($\gamma$):F($\gamma^3$)] $\leq$ 3

I have verified the inequality for a concrete case, but I'm not sure how to show that it is generally true. How can this be proven? Also, if we replace 3 by some other number, will analogous ...
1
vote
1answer
31 views

splitting field of $x^n-1$ over $\mathbb{Q}$

Is it true that the splitting field for $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q}(\xi_n)$ where $\xi_n$ is a primitive n$^{th}$ root of unity, making it an extension of degree $\phi(n)$ (Euler phi ...
0
votes
1answer
36 views

Stuck on last part of rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic?

I am checking to see if the rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic? I want to assume that the two rings are isomorphic and let $f$ be the isomorphism. I can let A = ...
1
vote
1answer
27 views

Splitting field for $x^4-x^2-2$

Am i right to say that the splitting field for $x^4-x^2-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2},i)$ which is of degree 4? i.e. $\{a+b\sqrt{2} + ci+di\sqrt{2} : a,b,c,d\in\mathbb{Q}\}$?
2
votes
1answer
17 views

Potential Frobenius automorphism question

Let $F$ be a finite field of characteristic $p$ of size $p^n$ for $n \ge 1$ with the base field $K \cong Z_p$. I'm attempting to prove that the map $\phi: F → F$ sending $u$ to $u^p$ for each $u \in ...
1
vote
1answer
40 views

homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ correspondence theorem question

I am looking at the homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ that sends $x$ to $1$. I need to explain what the Correspondence Theorem when applied to this map says about the ideals of ...
0
votes
0answers
20 views

[KL:L]<[K:K inter L] [duplicate]

I'm asked to find two extension fields of a field $F$, such that $K/F$ is normal, $L/F$ is algebraic and $[KL:L] < [K:K \cap L]$. The first part of the exercise says that if either $K$ or $L$ is ...
0
votes
1answer
28 views

Field of fractions of integral extension is an algebraic extension [duplicate]

Let $A\subset B$ be an integral extension. If $F$ and $E$ are the fields of fractions of $A$ and $B$, respectively, I want to show that $E$ is an algebraic extension of $F$. I know that since $A ...
0
votes
2answers
40 views

The splitting field of $x^{3}-2$ over $\mathbb{Q}$ and its degree.

The roots of $f = x^3 -2$ are $\{2^{1/3}, a, a^2\}$, where $a = \frac{-1+\sqrt{3}i}{2}$. So let $E$ be the splitting field of $f$ over $\mathbb{Q}$, then $E = \mathbb{Q}(2^{1/3}, a)$. Now I attempt ...
2
votes
1answer
49 views

$\mathbb{Z}_p$ necessarily realised as galois group of characteristic $p$ field?

Question I want to ask is practically precisely what's in the question but I will restate to make it clearer. Suppose $k$ is a field of characteristic $p$ which is not algebraically closed. Then we ...