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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$\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$

Let $\alpha$ be a zero of $f(x)=x^3+x+1 \in \mathbb{F_2}[x]$. Show that $\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$ So we need to show that $\mathbb{...
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2answers
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I don't understand this argument about a certain Galois group.

So I'm working with $\alpha = \sqrt{5+\sqrt{5}}$ and $E=\mathbb{Q}(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $f(x) = x^4 -10x^2 +20$ and I've determined that $E$ is its ...
3
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2answers
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Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
3
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2answers
57 views

Prove that $\mathbb{Q}$ has extensions of any finite degree in $\mathbb{C}$

This is a question from a course in Galois Theory and I am quite confused. In general, the degree of a field extension $E/F$ is the dimension of the vector space $E$. What would $E$ and $F$ be in ...
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Is $Q(2^{\frac14},\sqrt7)/Q(\sqrt2)$ normal?

I think it is because $(x^2-\sqrt2)(x^2-7)$ is a polynomial over $Q(\sqrt2)$ and $Q(2^{\frac14},\sqrt7)$ is the splitting field of this over $Q(\sqrt2)$ $\iff$ $Q(2^{\frac14},\sqrt7)$ is normal. So ...
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25 views

I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
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1answer
36 views

$G$ has normal subgroup of order 5

Let $L$ be the splitting field of $x^5-7$ over $Q$ and let $G=\text{Gal}(L/Q)$ (I) Prove that $G$ has a normal subgroup of order $5$ (II) Prove that $G$ has a subgroup of order $4$ that is not ...
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1answer
27 views

Find $[\mathbb Q(\sqrt3, \sqrt{5},2^{\frac13}):Q]$

$[Q(\sqrt3, \sqrt5:Q]=4$ $$ \begin{matrix} & & \mathbb Q(\sqrt3, \sqrt{5})(2^{\frac13}) & & \\ & \stackrel{a}{\diagup} & & \stackrel{b}{\diagdown} \\ \mathbb Q(\sqrt3, \...
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3answers
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How can I prove that $\mathbb Z_3[i]\cong \mathbb Z_3[x]/\langle x^2+1\rangle$?

Question: Let $\mathbb Z_{3}\left [ i \right ]=\left \{ a+bi\mid a,b \in \mathbb{Z}_{3} \right \}.$ Show that the field $\mathbb Z_{3}\left [ i \right ]$ is ring isomorphic to the field $\...
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1answer
65 views

Galois correspondence for the field extension $\mathbb{Q}(\omega_7)$

Let $E = \mathbb{Q}(\omega_7)$, where $\omega_7$ is the 7th root of unity. We know that $$\mathbb{Q}(\omega_7) \cong \mathbb{Z}_7^{\times},$$ where $\mathbb{Z}_7$ is the multiplicative group of units, ...
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0answers
37 views

the field of fractions of a domain $R$ does not have a projective cover if $R$ is not a field.

Let $R$ be an integer domain (so it commutes) that is not a field. Let $K$ be its field of fractions (also called localization). Prove that $K$ does not have a projective cover ( naturally viewed as ...
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1answer
67 views

Use Galois Theory to prove the existence of $A$ and $B$ such that $\mathbb{Q}(\sqrt{6+3\sqrt{3}})=\mathbb{Q}(\sqrt{A}, \sqrt{B})$

Use Galois Theory to prove the existence of $A$ and $B$ such that $\mathbb{Q}(\sqrt{6+3\sqrt{3}})=\mathbb{Q}(\sqrt{A}, \sqrt{B})$ So $\mathbb{Q}(\sqrt{6+3\sqrt{3}})$ is the field of rational numbers ...
2
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1answer
24 views

What does $ K^{\alpha}$ mean?

This is in context of a statement in galois theory: If $F \subseteq K \subseteq L$ and $K$ is splitting over $F$, then $K^{\alpha}=K$ for each $\alpha \in Aut(L)$.
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2answers
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Given the splitting field of a polynomial, how can I show that there are three intermediate extensions which aren't normal?

So $f = x^3 +9x -2$, and $E$ is its splitting field. I need to show that there are exactly three intermediate field extensions $K$ such that $\mathbb{Q} \subset K$ is not normal. By Descartes' rule ...
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3answers
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Finding a basis for a field

I have a polynomial f(x) = $x^3+x^2+1$ in $\mathbb{Z}_5[x]$ and it is given that F = $\mathbb{Z}_5[x]$/$<f(x)>$ = $\mathbb{Z}_5(\alpha)$ where $\alpha =x+<f(x)>$. I want to find a basis ...
2
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1answer
47 views

Show that $H = \mathbb{Z}_5[x]/\langle x^4+3x^3+x+4\rangle$ is not a field.

So I am looking over old exams in abstract algebra and I came across this question which seems to be a mistake. (Neither the original teacher who wrote it, nor my own teacher are available to answer) ...
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2answers
53 views

$f$ is factored into many same degree irreducible polynomials.

I met a problem when I study about Galois field and do this exercise. Hopefully, someone can help me. Suppose that $L/K$ is normal extension and $f$ is an irreducible polynomial in $K[X]$. Prove that ...
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1answer
22 views

Any ideal of a field $F$ is $0$ or $F$ itself

Prove that the only ideals of a field are $\left\{ 0 \right\}$ and the field itself. Let $F$ be a field and $I$ be an Ideal of $F$. Let $0 \ne x \in I$. Since $I$ is an Ideal of $F$, it is true ...
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0answers
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Galois Group Solution Check

Find the Galois group $G = \text{Gal}(\mathbb{Q}(\omega_{12})/\mathbb{Q})$, and its lattice of subgroups, where $\omega_{12}$ is the 12th root of unity. We have that $$\text{Gal}(\mathbb{Q}(\omega_{...
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What does $F[x]$ mean?

Lemma: $F$ is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ mean?
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Logic of Set Theory & Partially Order (Informative Discussion)

My final exam passed but, honestly I want to understand what this (Question 4) problem means because I don't know what it is asking for. I am a undergraduate, so it would be most helpful if the ...
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Prove $\pi+e$ or $\pi e$ is transcendental. [closed]

I understand to prove at least one of them irrational you would compose a function by which $\pi$ and $e$ are roots $((x-\pi)(x-e))$, and show that at least one coefficient cannot be rational because $...
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2answers
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Is $Q(5^{1/4},√11,i)/Q $ normal

I think all roots of $x^4-5$ and $x^2-11$ and $x^2+1$ are in the field but it seems impossible to find a irreducible polynomial that contains all those roots. How can we check if it is normal? I ...
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1answer
15 views

Is $Q( 5^{1/3},√7)/Q$ a normal extension?

Can someone give me a working out of this please. I don't really have any detailed examples in my notes so i have no idea about this normal extension stuff. I do know that an extension $K\subseteq L$ ...
2
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2answers
28 views

Finding subfields of degree $3$

Let $F$ be the splitting field of the polynomial $X^3−5$ over $Q$. Let $G = Gal(F/Q)$. (i) Determine $G$ up to isomorphism. (ii) Find all subfields $M$ of $F$ such that $Q⊆M$ and $[M :Q] = 3$. [Give ...
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1answer
28 views

Find $[Q(w,√3) :Q] $ and find a basis

Let $w∈\mathbb C$ be a root of the polynomial $X^4−12$. Determine $[Q(w,√3) :Q]$ and give a basis of $Q(w,√3)$ over $Q$. [You may express the elements of your basis in terms of w. Note that the exact ...
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1answer
45 views

Zeros of specialization of a family of polynomials [closed]

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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2answers
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Is $\mathbb Q(\sqrt{4+i\sqrt{20}},\sqrt{4-i\sqrt{20}})=\mathbb Q (\sqrt{4+i\sqrt{20}})$?

I don't know if this is true but it is trivial that the right is contained in the left. With the other inclusion, I think the only non trivial thing to check is if $A=\sqrt{4-i\sqrt{20}} \in \mathbb Q ...
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0answers
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Normal extension of a field

Let $F$ be an extension of $K$ (they are both fields). I know that if $F$ has finite degree over $K$, then the following things are equivalent: 1) $F$ is such that every irreducible polynomial in $K[...
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0answers
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Determine $\text{Gal}(L/Q)$ and its action on a basis of $L$.

(a) Find the minimal polynomial $f$ of $√5+i$ over $Q$. (b) Find the splitting field $L$ of this $f$. (c) Determine $\text{Gal}(L/Q)$ and its action on a basis of $L$. Stuck on part c. For (a) it ...
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1answer
24 views

$[F(a,b):F(a)]=[F(b):F]\iff F(a)\cap F(b)=F$

Let $F$ be a field and $a,b$ be elements of some algebraic extension of $F$. Is it true that $[F(a,b):F(a)]=[F(b):F]\iff F(a)\cap F(b)=F$? I have a proof for the forward implication: Let $c\in F(a)\...
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2answers
32 views

Find the minimal polynomial $f$ of $√5+i$ over $\mathbb Q$

The candidate is $x^4-8x^2+36=0$ but we cant use Eisenstein here to prove irreducibility. What do we do?
2
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2answers
49 views

How is $[Q(\sqrt2, \sqrt3 ) : Q(\sqrt2)]=2$?

$\mathbb{Q}$ is the rationals. I know that $\sqrt3 \notin \mathbb{Q}(\sqrt2)$ but so what? The answer to this question seems to be based upon that. Really don't understand what that means in finding ...
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Galois group of extension generated by all cubic roots

Let $K/\mathbb{Q}$ be generated by all cubic roots of rational numbers, that is $K=\mathbb{Q}(\{\sqrt[3]{a}:a\in\mathbb{Q}\})$. I would like to understand its Galois group. I only could prove that $...
0
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1answer
32 views

How do I find the intermediate field extensions $F \subset K \subset E$ when ${\rm Gal}(E/F)$ is known?

I have the polynomial $x^3 -2 \in \mathbb{Q}[x]$, of which the splitting field is $E = \mathbb{Q}(2^{1/3}, \omega)$ where $\omega = e^{2\pi i/3}$. The Galois group of $E$ over $\mathbb{Q}$ is ...
2
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1answer
48 views

Determine $[F(a):F]$ if $a\in K$ has $k$ distinct images under Galois group

Suppose that $K/F$ is Galois extension and $a \in K$ has exactly $k$ distinct images under $Gal(K/F)$. Show that $[F(a):F]=k$. My guess is that the images of $a$ form a basis of $F(a)$ over $F$. But ...
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0answers
36 views

Let K be a field, $b\in K$, does there exist a field L and $c\in L$ with: there is no homomorphism, such as $\phi(b)=c$?

I want to show that for a given field $\mathbf{K}$ with chracteristic zero and a fixed element $b\in \mathbf{K}$ there exists another field $\mathbf{L}$ with characteristic zero and an element $c\in \...
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1answer
31 views

What can we say about the dimension of $[ E : F]$ if $F finite$ and $f \in F[x]$ min. polynomial [closed]

suppose I have a field $F$ and $\alpha \notin F$. $F$ is finite so $char(F)$ is some $p \in \mathbb{N}$ when I have a minimal polynomial $f_\alpha \in F[x]$ with $deg(f_\alpha)=n$ then the dimension ...
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1answer
66 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates of $\...
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1answer
47 views

For $\alpha = \sqrt{2+\sqrt{2}}$, what is ${\rm Gal}(\mathbb{Q}(\alpha)/\mathbb{Q}$?

So $\alpha = \sqrt{2+\sqrt{2}}$, and I've already found the minimal polynomial of $\alpha$ over $\mathbb{Q}$ to be $p=x^4 - 4x^2 + 2$ and shown that $\mathbb{Q}(\alpha)$ is a normal extension. Now I ...
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1answer
29 views

Curious about field extensions and the relationship between the automorphisms of the fields' Galois groups

Let's say we have field extensions $F(\alpha)$, $F(\beta)$ and $F(\alpha, \beta)$of a field $F$. My question is this: if $\sigma \in {\rm Gal}(F(\alpha)/F)$, is it correct to say that $\sigma \in {\rm ...
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elements of a finite field vs its characteristic

What is the difference saying, '' a finite field with $q$ elements'' or ''a field with characteristic $p$'' ? a finite field must be of prime characteristic and vice versa, what is the difference ...
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1answer
155 views

Is ({1, 0}, ⊕, ∨) a field? and Is ({1, 0}, ⊕, ∧) a field?

1 and 0 denote the logical statements True and False. These two questions are for homework so would rather an answer that could help explain it to me then just a straight answer. Thanks to anyone who ...
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2answers
40 views

Example of Galois extension over Q which is not cyclotomic

So, prof. introduced Galois extensions yesterday and I do apologise if I did not get something correctly. So, if I am right every finite extension of finite field is almost obviously Galois(using ...
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1answer
22 views

Splitting field and automorphisms

I know that if $K$ is a field and $f\in K[x]$, then there exists a splitting field of $f$ on $K$. If one has two isomorphic fields $K_1$ and $K_2$ (say $\sigma$ an isomorphism) and $f\in K_1[x]$, ...
4
votes
1answer
154 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
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0answers
37 views

Galois Group is $\mathbb{Z}/4\mathbb{Z}$.

Let $K \subseteq L$ be a Galois field extension with Gal$(L/K) \cong \mathbb{Z}/4\mathbb{Z}$. Show that $L$ is the splitting field of a polynomial $f(x)=(x^2 −a)^2 −b$ for elements $a,b \in K$ such ...
0
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0answers
45 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
0
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0answers
23 views

Galois Group of Splitting Field, $S_4$

I've shown that the polynomial $x^4+px+p \in \mathbb{Q}[x]$, where $p$ is prime, is irreducible by Eisenstein's criterion. However, it remains to be shown that the Galois group of the splitting field ...
1
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1answer
48 views

Galois Extension.

Suppose K is a finite field extension of $\mathbb{Q}$. Let K ⊆ L be a Galois field extension and K ⊆ K′ be a finite field extension. Show that K′ ⊆ K′L is a Galois field extension and $$\text{Gal}(K′L ...