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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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 ...
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63 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 ...
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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 ...
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2answers
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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 ...
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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= ...
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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
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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 ...
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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 ...
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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 $\beta = \alpha$, ...
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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
21 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 ...
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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 ...
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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
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1answer
30 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
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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 = ...
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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. $\{a+b\sqrt{2} + ci+di\sqrt{2} : a,b,c,d\in\mathbb{Q}\}$?
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1answer
16 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 ...
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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 ...
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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
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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 ...
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2answers
39 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
44 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 ...
2
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0answers
19 views

Automorphisms of field generated by two coprime elements

I would like to know if the follwing statement is true: Let $F$ be a field and let $a,b$ be algebraic over $F$ with coprime degrees $m$ and $n$, respectively. Suppose $F(b)/F$ is normal. Letting $K = ...
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1answer
27 views

Field automorphisms of extension generated by two coprime algebraic elements.

Let $F$ be a field and let $a,b$ be algebraic over $F$ with $[F(a) : F] = n$ and $[F(b) : F] = m$ coprime. Let $\sigma \in \textrm{Aut}(F(a,b)/F)$. Is it true that $\sigma(F(a)) = F(a)$ and ...
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1answer
50 views

I know Galois theory is used to study fields using properties of groups. Is it ever used to study groups using properties of fields?

More specifically, are there any results in pure, abstract group theory that are most easily proved using Galois theory?
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35 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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3answers
42 views

How can it be shown that for some prime p, $\mathbb{Q}$[$\sqrt{p}$, $\sqrt[3]{p}$] = $\mathbb{Q}$[$\sqrt[6]{p}$]?

I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here.
2
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0answers
20 views

On the restriction of field homomorphisms to subfields

Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace ...
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3answers
32 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
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2answers
42 views

how can one show that $\mathbb{Q}$($\sqrt{3}$, $\sqrt[3]{3}$, $\sqrt[4]{3}$, …) is algebraic but not finite dimensional?

The fact that this extension is infinite seems almost obvious and this is what makes it difficult to prove that the extension is algebraic. I would be able to do it for a finite case by identifying ...
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0answers
19 views

How to calculate the discriminant of a cubic equation easily

I'm trying to show the degree of the splitting field of a cubic polynomial with a zero quadratic term is related to the discriminant of the polynomial. On this process, i am trying to find the product ...
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1answer
31 views

simple algebraic extensions with the same minimal polynomial [closed]

I can't see why $ji^{-1}$ is the identity on $K$, could someone explain please
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1answer
26 views

Finding a transcendence basis of a field over a rational function field in positive characteristic

Let $p$ be a prime number, and $k = \Bbb{F}_p(t)$ be a function field over $\Bbb{F}_p$. Let $R = k[x,y]/(x^p+y^p-t)$, and $K = \operatorname{QF}(R)$ be the quotient field of $R$. I need to find the ...
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1answer
39 views

How to determine the minimal polynomial of $\sqrt{3 + 2\sqrt{2}}$ over $\mathbb{Q}$?

I first let $\alpha = \sqrt{3 + 2\sqrt{2}}$ and $\alpha^2 - 3 = 2\sqrt{2}$. This gives us $(\alpha^3 - 2)^2 = 8$. Expand the polynomial we obtain that $x^4 - 6x^2 +1$ has $\sqrt{3 + 2\sqrt{2}}$ as a ...
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0answers
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Quick help on why this extension is of degree $2$

The set up is, Splitting field $K=\mathbb{Q}(i,\alpha)$ where $\alpha=2^{\frac{1}{4}} \in \mathbb{R}$. The three obvious subfields of $K$ with degree $2$ over $\mathbb{Q}$ are..? Answer is ...
2
votes
0answers
21 views

Maximum ideal in field

Let $k$ be a field, $n ∈ \mathbb{Z}>0$, and $α_1, α_2, ..., α_n ∈ k$. Prove: $(x_1 − α_1, x_2 − α_2, \ldots, x_n − α_n)$ is a maximal ideal. I cannot figure out how to prove this; what is meant ...
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0answers
29 views

Why does a subfield of $\mathbb{R}$ not contain every real number?

Let $F$ be a subfield of $\mathbb{R}$. Since the completeness axiom holds in $\mathbb{R}$, it will also hold in $F$. But every real number can be represented as an infinite cauchy series of rational ...
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2answers
43 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
2
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1answer
53 views

Any method to solve this system of equation?

We have m variables $ x_{1},x_{2},...,x_{m} $ which are elements of field $F_{p}$ and we are given m equations of the form $$\sum_{i=1}^{m} x_{i}^{n} = c_{n} \mod p \qquad for \: 1 \le n \le m$$ ...
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1answer
37 views

Show $\sqrt[3]{2}\notin F$

Suppose that $F$ is the infinite extension of $\mathbb{Q}$ obtained by adjoining the square root of every integer (positive or negative). I'm trying to show that $\sqrt[3]{2}\notin F$. I have no idea ...
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1answer
39 views

Is $K^{n}$ Zariski Hausdorff when $K$ is a finite field?

Assume that $K$ is a finite field. Is it true to say that $K^{n}$ is a Hausdorff topological space with Zariski topology?
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1answer
30 views

How do I find a basis of $\mathbb{Q}(i,\sqrt{2}+i,\sqrt{3}+i)$ over $\mathbb{Q}$?

I can see that $[\mathbb{Q}(i):\mathbb{Q}]=2$ and that each of $[\mathbb{Q}(\sqrt{2}+i):\mathbb{Q}]$ and $[\mathbb{Q}(\sqrt{3}+i):\mathbb{Q}]$ is $4$. This implies that ...
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1answer
32 views

structure of extension by quadratic elements

Is it true that if $\sqrt{b}\not\in\mathbb{Q}(\sqrt{a})$ for $a,b$ not squares in $\mathbb{Q}$, then $Gal(\mathbb{Q}(\sqrt{a},\sqrt{b}))\cong Gal(\mathbb{Q}(\sqrt{2},\sqrt{3}))$? Im seeing a bunch of ...
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0answers
25 views

Galois group of degree 3 extension

For $d$ not a perfect cube in $\mathbb{Q}$, I want to compute $Gal(\mathbb{Q}(\sqrt[3]{d})/\mathbb{Q})$. I believe it has to be the trivial group, since it sends roots of the minimal polynomial ...
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1answer
27 views

Show that $\mathbb{Q}(\sqrt{2})$ is a field.

Proof: Since $\mathbb{Q}$ is a field, then $\mathbb{Q}$ is a domain. (Theorem: if $R$ is a domain, then $R[x]$ is a field.) By the theorem, $\mathbb{Q}[x]$ is a field. So, letting $x = \sqrt{2}$, ...
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1answer
32 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
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votes
1answer
39 views

The Ring extension isomorphic to the field extension

Let $\alpha$ be algebraic over $F$, with $F(\alpha)$ the smallest field containing both $F$ and $\alpha$, and with $F[\alpha]$ the smallest ring containing both $F$ and $\alpha$. I want to show ...
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0answers
24 views

Center of matrices over a field [duplicate]

I'm trying to find the center of $\mathbb{M}_n(K)$ with $K$ a field. I know what the center would be if $K$ was a ring, but I think this isn't the same for a field $K$. In particular I'm trying to ...
0
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1answer
37 views

How to find the minimal polynomial of $\mathbb{Q}(3^{1/2}+5^{1/3})$ over $\mathbb{Q}$?

How would I find this polynomial algebraically? For example, if I wanted to find the minimal polynomial of $\mathbb{Q}(3^{1/2})$ over $\mathbb{Q}$, I would set $\alpha=3^{1/2}$, square both sides, ...
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0answers
16 views

Factoring Bivariate Polynomial which Vanishes on Curve

Let $K$ be an infinite field, and let $f \in K[x,y]$. Suppose $f$ vanishes on $x = y$. Show that $x-y$ divides $f$. I want to be able to use the division algorithm, so I first form the field of ...