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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Primitive elements for $K=\Bbb{Q}(\sqrt{2},\sqrt{3})$

The key lemma for proving the primitive Element Theorem (for finite extension of a field $F$ with characteristic $0$) in Artin's Algebra (2nd edition) is the following: Suppose $char F=0$ and ...
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+50

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
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Show that any finite extension of $\mathbb{Q}$ is not algebraically closed.

EDIT: Entirely wrong question. I wanted to ask something else. How do I show that any finite extension of $\mathbb{Q}$ is not algebraically closed. In other words, the algebraic closure of ...
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3answers
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Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$

Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F[x].$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$ Please help me. I'm absolutely clueless.
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Fields with Additive identity powers

Would it be possible to have a field (or field-like structure) with an additive identity $k$ where $k^a\neq k^b$ for $a\neq b$? I need this because I'm working with a field-like structure where if I ...
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1answer
49 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
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1answer
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Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...
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4answers
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How to show that $\mathbb Q(\sqrt 2)$ is not field isomorphic to $\mathbb Q(\sqrt 3).$

How to show that $\mathbb Q(\sqrt 2)$ is not field isomorphic to $\mathbb Q(\sqrt 3)?$ My text provides the hint as: Any isomorphism from $\mathbb Q(\sqrt 2)\to\mathbb Q(\sqrt 3)$ is identity when ...
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1answer
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Express $a^5$ in terms of $c_0+c_1a+c_2a^2.$

Let $F=\mathbb Z_2,f(x)=x^3+x+1\in F[x].$ Suppose $a$ is a zero of $f(x)$ in some extension of $F.$ Then $F(a)\simeq F[x]/\langle f(x)\rangle$ and there is an isomorphism $$\phi:F[x]/\langle ...
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1answer
16 views

In algebraic extension, field homomorphism induces isomorphism.

I read this page's first answer. But I'm curious about why $\varphi$ induces injective map $S \to S$. Isn't it possible to make $\varphi(\alpha)= k$ such that $k$ is not root of $f(X)$?
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1answer
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Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
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0answers
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Field Extension of Rational Functions

Let $L = F(x)$ be the field of rational functions over a field F. Let $u \in L \backslash F$. Let $K = F(u)$. If u can be written as $\frac{f}{g}$ where $gcd(f,g) = 1$, then prove $[L:K]$ = max {deg ...
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1answer
59 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
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1answer
39 views

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$?

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$? I think that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}):\mathbb{Q}] = 8$, but not really sure how to ...
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2answers
55 views

Why must a field with a cyclic group of units be finite?

Let $F$ be a field and $F^* \subseteq F$ be its group of units. If $F^*$ is cyclic show that $F$ is finite. I'm a bit stuck. I know that I can represent $F^* = \langle u \rangle$ for some $u \in F^*$ ...
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1answer
79 views

Is $x^4+2$ irreducible over $\Bbb{Q}(i)$?

Let $f(x)=x^4+2$. Using the Eisenstein test to $f(x+2)$, one can show that $f$ is irreducible over ${\Bbb Q}$. Let $\beta$ be a complex root of $f$. Then the question in the title is equivalent to ...
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1answer
22 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
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3answers
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Characterization of a subfield $K \varsubsetneq \mathbb {C}$ and $x\in \mathbb{R}$

Characterize $x \in \mathbb R$ such that : There exist a subfield $K \varsubsetneq \mathbb C$ such that $K(x) = \mathbb C$ -All subfields $K$ of $\mathbb{C}$ contain $\mathbb Q$, then all $x\notin ...
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Construction of the field of real numbers within $ZF$ [duplicate]

I am interested in a problem whether the field of real numbers can be constructed within $ZF$. I will state the problem more precisely as follows. Definition 1 An ordered field $K$ is called ...
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1answer
30 views

What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in ...
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1answer
53 views

Basis for $\mathbb{Q}[\sqrt{8}]$ over $\mathbb{Q}[\sqrt{2}]$

Provided that $x^2-8$ is the minimal polynomial for $\mathbb Q[\sqrt8]$ and $x^2-2$ is minimal for $\mathbb Q[\sqrt 2]$ we should have a basis with four elements. Thus far I know $1$ and $\sqrt 2$ ...
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Dummit and Foote page 526

I'm having trouble with a line of example 2 on page 526. Consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$. generated over $\mathbb{Q}$ by $\sqrt{2}$ and $\sqrt{3}$. Since $\sqrt{3}$ is of degree ...
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2answers
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Is the algebraic subextension of a finitely generated field extension finitely generated?

This question is motivated by this other question (and its answer). Suppose we have a field $F$, possibly imperfect. Consider the finitely generated field extension $F(a_1,\ldots,a_n)$. Is it always ...
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21 views

About irreducibility over Fields of non-zero characteristic. [closed]

Question-1: Let $F$ be an imperfect field with characteristic $p$. Then $(x^{p^n}) - a$ is irreducible for $a$ in $F^p$ and $n=0,1,2,\ldots$ Question-2: Let $f(x)$ be irreducible in $F[x]$, $F$ of ...
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3answers
71 views

Is is true that $\zeta$ has finite order?

Let $\zeta$ be a complex number on the unit circle $\{z\in \mathbb{C}: |z|=1\}$.Suppose that $[\mathbb{Q}(\zeta):\mathbb{Q}] < \infty$.Is it true that $\zeta ^n=1$ for some positive integer $n$?
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1answer
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Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
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Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
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1answer
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$\Bbb Q(a, \sqrt{\vartriangle(f)})$ is a splitting field in this case.

Let $f\in \Bbb Q [x]$ be monic irreducible and of degree 3. Prove that if $\alpha \in \Bbb C$ is any zero of $f$ then the field $\Bbb Q(\alpha, \sqrt{\vartriangle(f)}) \subset \Bbb C$ , is a splitting ...
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1answer
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Binomial formula over an arbitrary field

I'm working on a problem (namely, if $\alpha + \beta$ is algebraic over $F$ then $\alpha$ is algebraic over $F[\beta]$), and the binomial formula appeared. For the problem, I used the fact that, for ...
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1answer
59 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
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1answer
27 views

Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
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2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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1answer
37 views

Uncountable, algebraically independent subset of $\mathbb{C}$?

Does such a subset exist? I am interested in algebraic independence over $\mathbb{Q}$. Could this be proven in an abstract way or would it be more appropriate to construct an explicit example?
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Equal fields and dimension

Let $K \subseteq L$ be fields. Show that K=L if and only if $dim_k L=1$ I know that I will have to show both directions of the implication. I'm very new to the topic of fields so I'm still ...
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1answer
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Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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5answers
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How can one find a set of given cardinality and disjoint from a given set?

In Algebra by Serge Lang, the author asserts, to prove the existence of a field extension where an irreducible polynom has a root, that if you take one set $A$ and a cardinal $\mathcal{C}$, that you ...
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countable set that contains 1 and pi and has polynomial with coefficients in set s.t. all real roots are in set

Deduce that there is a countable set X that contains the real numbers 1 and pi and has the further property that if P is any non-zero polynomial with coefficients in X, then all real roots of P belong ...
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3answers
43 views

An irreducible polynomial cannot share a root with a polynomial without dividing it

There is a lemma of Galois stating, "An irreducible equation can have no common root with a rational equation without dividing it". His definitions are a little bit imprecise, but I think he means: ...
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Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
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The Galois closure

If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
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Capelli Lemma for polynomials

I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof. It states like this: Let $K$ be ...
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1answer
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Proving the Galois Group of an extension is abelian

Let $E_{1}, E_{2}$ be subfields of $\mathbb{C}$. Suppose $E_{1}|\mathbb{Q}$ and $E_{2}|\mathbb{Q}$ are finite Galois extensions and $G(E_{1}:\mathbb{Q})\cong$ $\mathbb{Z}_{6}\cong$ ...
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A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
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14 views

Field Extensions Identities

I'm working on proving some identities but I need some help clarifying the notation and what exactly each statement is saying. Prove the following identities. (a) $K(A) = QF (K[A])$ (b) $R[A_1 ...
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How many elements does $\mathbb F = \mathbb Z_{7}[x]/I$ contain?

Let $p(x) \in \mathbb Z_{7}[x]$, given by $p(x) = x^{2}+3x+1$ and let $I = <p(x)>$ be the ideal in $\mathbb Z_{7}[x]$ constructed by $p(x)$. How many elements does $\mathbb F = \mathbb ...
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1answer
34 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
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Splitting Fields Proof

Let $K\subseteq L$ be fields and $ f \in K[X] \setminus K$. Prove that the following are equivalent. (a) $L$ is a splitting field for $f$ over $K$ (b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ ...
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1answer
37 views

If an identity in the language of rings holds for all fields, does it necessarily hold for all commutative rings?

It is weirdly difficult to find new identities for ring theory (other than commutativity) that make it more like field theory. This motivates my: Question. If an identity in the language of rings ...
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Let $\mathbb F = \mathbb Z_{3}[x]/\langle x^{2}+1 \rangle$ Show that $\phi : G \to \mathbb F\backslash\{0\}$ defined by …

Let $G = \left\{ \begin{bmatrix} \alpha & \beta \\ 2\beta & \alpha \end{bmatrix} \;\Bigg| \; \alpha,\beta \in \mathbb Z_{3} (\alpha,\beta) \neq (0,0) \right\}$ Let $\mathbb F = \mathbb ...
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1answer
36 views

$x^4-3x^2+4$ irreducible over over $\mathbb{Q}$

I need to prove irreducibility of $x^4-3x^2+4$ over $\mathbb{Q}$. It can't have any linear factor since it doesn't have any root in $\mathbb{Q}$ because any $\alpha \in \mathbb{Q}$ is a root only ...