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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Galois group of a polynomial and subfields

Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible quartic whose splitting field has Galois group $S_4$ over $\mathbb{Q}$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$. a) Prove ...
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1answer
15 views

This is only a subspace if $b=0$ - Axler - LADR p13

I have written here in Axler - Linear Algebra Done Right, page $13$. If $b\in \mathbb{F}$, then $\{(x_1,x_2,x_3,x_4)\in \mathbb{F}^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb{F}^4$ if and only ...
2
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2answers
28 views

Show that an extension is separable

Let $K$ be a field with $\operatorname{char} K=p$, where $p$ is a prime, and let the degree of the extension $K \leq L$ be coprime to $p$. How can I show that the extension is separable?? Could you ...
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1answer
16 views

Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

I want to prove, that $\Bbb F_p^\times = MRP(p)$. I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$ But I do not know how to continue this idea.
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2answers
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Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$. Assume polynomial $p(x)\in F[x]$ s.t. $p(r^2)=0$ If $r\in K$ and $r^2$ is ...
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3answers
376 views

Each element is a square of some element

I have to show that each element of $\mathbb{F}_{2^n}$ is a square of some element. Could you give me some hints how I could do that??
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0answers
48 views

show that $\mathbf Q(\sqrt2,\mathrm i)=\mathbf Q( \sqrt2+\mathrm i)$.

show that $\mathbf Q(\sqrt2,\mathrm i)=\mathbf Q( \sqrt2+\mathrm i)$. I have never seen the $\mathbf Q(\sqrt2, \mathrm i)$ notation before so I am confused as to what it means - is it field generated ...
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0answers
30 views

Automorphisms of the field of real numbers

How many automorphisms of the field of real numbers are there? If we use only algebra not analysis (analysis means every real number is a limit of some sequence of rational numbers and algebra means ...
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votes
2answers
38 views

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$ I really have no idea how to prove this problem. I started to consider: Assume $\sqrt[m]{p}\in ...
2
votes
2answers
24 views

Dimension Field True/False.

I'm having trouble approaching how to determine truthfulness and falsehood of the following type of problems. $F$ and $K$ are fields. 1) Suppose that $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ ...
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0answers
31 views

Subfield Criteria - Proof or Counterexample

I am interested in whether the following claim is true for all fields $F$: Conjecture: A subset $X\subset F$ is a subfield if and only if (1) $1\in X$, (2) $x,y\in X\Rightarrow x-y\in X$; and (3) ...
2
votes
1answer
60 views

Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
0
votes
0answers
16 views

Show that it is/is not a normal extension

Let $a \in \mathbb{R}$ with $a^4=5$. Show that: $\mathbb{Q}(ia^2)$ is a normal extension of $\mathbb{Q}$. $\mathbb{Q}(a+ia)$ is a normal extension of $\mathbb{Q}(ia^2)$ $\mathbb{Q}(a+ia)$ is not a ...
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1answer
56 views

is $K = \mathbb{Z}[x] / (x^4+9x+6)$ a field or not?

Let $p(x)=x^4+9x+6$. it is irreducible over $\mathbb{Z}[x]$ by Eisenstein criterion(Because $3^2$ does not divide 6). My question is whether $K = \mathbb{Z}[x] / (p(x))$ is a field or not ? There is ...
17
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2answers
1k views

Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
2
votes
2answers
67 views

Show that $\mathbb{Q}( \sqrt2) \neq \mathbb{Q}( \sqrt3)$

The way that I'm thinking is by showing that the field extension $\mathbb{Q}( \sqrt2) /( \sqrt3) \neq \mathbb{Q}( \sqrt3)$, but is there a simpler way I'm ignoring?
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1answer
15 views

Notation question regarding field extensions (What does $K^2 \subseteq k$ mean)

recently I am reading a paper on pfister forms in characteristic 2 and stumbled across a notation I do not know. It can be found here Suppose $k$ is an arbitrary field of characteristic 2. Let ...
2
votes
1answer
18 views

Do two isomorphic finite field extensions have the same dimension?

If $E = F(u_1, \cdots u_n) \cong \bar{E} = F(v_1, \cdots v_m)$ then do the two extensions necessarily have the same dimension over $F$?
2
votes
1answer
36 views

Questions on the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$

Given the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$ of $\mathbb{Q}$, and letting $a = [x] ∈ K$; 1) Show $K ≃ \mathbb{Q}(\sqrt5) $ and $[K : \mathbb{Q}] = 2.$ 2)Find the ...
2
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3answers
56 views

Explain to me the difference between the notation $\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$

Please explain to me the difference between the notation $\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$. I know that these two fields are equal. But what difference do the different brackets imply? ...
3
votes
1answer
41 views

Universal property of the algebraic closure of a field

At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books. Exercise 1.2a is then to convince ...
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votes
0answers
15 views

Normal basis in every subgroup

Let $K$ be finite Galois extension of $k$, $G=Gal(K/k)$ and $k$ infinite. Prove: Exists $\beta\in K$ such that for every $N<G$, we have $\{\sigma(\beta):\ \sigma\in N\}$ is a basis for ...
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votes
2answers
19 views

The field closure of a countable union of countable fields is countable?

If $ K_1 \subset K_2\cdots \subset K_n \subset \cdots$ is a tower of countable fields then their union $ \bigcup_n K_n$ is a countable field. If $\{K_a\}$ is a countable family, but not a tower, ...
2
votes
0answers
32 views

element in field, not a square [duplicate]

I am doing a specific exercise where the quaternion group is realised as a Galois group of some field extension. It goes like this: let $K = \mathbb Q(\sqrt 2,\sqrt 3)$ and $\alpha = (2 +\sqrt 2)(3 ...
0
votes
1answer
19 views

Are simple extensions of isomorphic fields isomorphic?

If $F(u) \cong F(v)$, and both are subfields of a larger extension, $K$, then for $k \in K$, is $F(u)(k) \cong F(v)(k)$?
0
votes
1answer
30 views

Field, algebraic element

1) Let $E/F$ an extension and let $\alpha,\beta\in E$ be algebraic elements over $F$. If $\alpha\neq 0$, prove that $\alpha+\beta$, $\alpha\beta$ and $\alpha^{-1}$ are all algebraic over $F$. 2) If ...
0
votes
0answers
13 views

Trace Map and Solution to Equation

Consider the equation $x^{31} - 1 =0$. Determine the number of solutions $\gamma \in \mathbb{F}_{1024}$ to the equation that satisfies $Tr(\gamma)=0$. I did some research on properties of field ...
0
votes
1answer
24 views

Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...
3
votes
0answers
74 views

What branch of math is this?

In this paper: http://arxiv.org/pdf/hep-th/0505016v1.pdf what are the branch(es) of math being used? The unnumbered eq. on the top of page 3 and eq. (7) are good examples. All I've been able to figure ...
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2answers
41 views

Let k be a field and let p, q ∈ N be two prime numbers such that p · 1 = q · 1 = 0. Show that p = q.

My current train of though is letting p =/= q then proving that q must be divisible by p, the contradiction then being that q is prime. But I'm not sure how to go about doing this.
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1answer
39 views

Help with problem of the book Algebra of T. Hungerford

This is a problem in “Algebra” by T. Hungerford: If $|K|=q$ and $f\in K[x]$ is irreducible, then $f$ divides $x^{q^n}-x$ if and only if $\deg f$ divides $n$. I found it difficult to solve ...
0
votes
1answer
29 views

Show that the Frobenius homomophism is bijective.

Let k be a finite field with characteristic $p ≥ 2$, show that the homomorphism $F:k\rightarrow k$ where $x \mapsto x^p$ is bijective. Could someone please explain the statement $p ≥ 2$ to me, as ...
0
votes
1answer
25 views

Showing two field extensions (of Q) are isomorphic using primitive elements, and why every element is primitive

I'm taking a class on Galois Theory in another language and the prof is saying my answer on this is incorrect and I'm wondering why, particularly since sometimes there's a language barrier. Basically ...
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votes
2answers
39 views

Is $x^2-2$ irreducible over R and Q?

I'm not sure if it is valid to say that $x^2 - 2$ can be factorised to $2\cdot\left(\frac 12x^2 - 1\right)$ for it to be reducible in Q. Though I know $(x + \sqrt{2})(x - \sqrt{2})$ works in the ...
0
votes
1answer
27 views

Show that $x^3 + 3x^2 + 9x + 3$ and $x^3 + 3x^2 + 3x − 4$ are irreducible in $\mathbb{Z}[x]$

I need to show, as stated in the title, that $x^3 + 3x^2 + 9x + 3$ and $x^3 + 3x^2 + 3x − 4$ are irreducible in $\mathbb{Z}[x]$ I know that in case of second polynomial, if $f(x-1) = x^3 - 5$ which ...
0
votes
1answer
22 views

Algebraic closure.

I was trying to solve this: Let $E$ be a extension field of $K$. If $K$ is a algebraically closed field, then the algebraic closure $A$ of $K$ on $E$ is a algebraically closed field. But for ...
1
vote
1answer
19 views

Ruler and compass constructions and fields

Use the fact that $\alpha=$cos$(2\pi/5)$ satisfies the equation $x^2+x-1=0$ to conclude that the regular $5$-gon is constructible by straightedge and compass. So, the polynomial $x^2+x-1$ is ...
3
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1answer
30 views

Showing that a polynomial over subring is reducible

Suppose that $R_1\subseteq R_2$, and both are integral domains. Further suppose that $R_2$ is a field, where each element $r\in R_2$ is a zero of a polynomial in $R_1[x]$ with the leading coefficient ...
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votes
1answer
20 views

quadratic equation modulo some number

I read a post that $$ax^2+bx+c \equiv 1 \pmod p$$ can be solved in a similar way we solve a simple quadratic equation, just by replacing division by $2a$ by modulo inverse of $2a$ and square root of ...
1
vote
1answer
25 views

How to find the splitting field of $X^4-10X^2+1$?

How to find the splitting field of $X^4-10X^2+1$ ? I found the roots \begin{align*} X^4-10X^2+1=0&\iff (X^2-5)^2-24=0\\ &\iff X^2-5=\pm 2\sqrt 6\\ &\iff X^2=5\pm 2\sqrt 6\\ &\iff ...
3
votes
2answers
44 views

The splitting field of $x^4-4$

I have to find the splitting field of $x^4-4$. I say that $$x^2-4=(x^2-2)(x^2+2)=(x-\sqrt 2)(x+\sqrt 2)(x-i\sqrt 2)(x+i\sqrt 2)$$ then I would say that the splitting field is given by $E=\mathbb ...
0
votes
2answers
28 views

Diagonalisable matrices over different fields

I believe this fits in with my knowledge of Jordan Normal form, however I am not sure how to approach the question itself? I am especially lost with $F_7$
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3answers
41 views

As fields $F$ not isomorphic with $\mathbb R$ , but as sets $F \sim \mathbb R$ , example ? [closed]

I am looking for an example of a field $(F,+,.)$ such that as fields , $F$ , $\mathbb R$ are not isomorphic but there exist a bijection between $F$ and $\mathbb R$ .
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1answer
45 views

Example of infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ? [duplicate]

It is known that any infinite cyclic group can never be a vector space , from this we can derive that if $(F,+,.)$ is an infinite field then $(F,+)$ cannot be cyclic . I am asking , is there any ...
1
vote
1answer
28 views

Normal field extension implies splitting field

I feel like this fact should be easy but I'm struggling to see it. If I have a polynomial $f \in K[x]$ which is irreducible and has roots $\alpha$, $\beta$ in some finite normal (over $K$) extension ...
1
vote
1answer
30 views

Calculating fixed fields

I have to show that $\mathbb{Q}(\zeta)^{<\sigma>}$ $=$ $\mathbb{Q}(\zeta + \frac{1}{\zeta})$ $\sigma \colon L \to L$ is defined by $\sigma(\alpha) = \overline{\alpha}$, where ...
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0answers
17 views

Galois group is $S_{n}$

If $K/F$ is Galois extension with Galois group $S_{n}$ then show that $K$ is the splitting field of a degree $n$ polynomial irreducible over $F$. We know $K$ is splitting field of some separable ...
1
vote
1answer
43 views

Degree 4 extension of $\mathbb {Q}$ with no intermediate field

I am looking for a degree $4$ extension of $\mathbb {Q}$ with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible ...
1
vote
1answer
35 views

Can every element of an algebraic field extension, $E \subseteq F$ be represented as $f(s)$, where $s$ is another member and $f(x) \in F[x]$

Specifically, could every member of $E$ be generated by one element $s$, by evaluating different polynomials of $s$ with coefficients in $F$?
0
votes
2answers
24 views

Field extensions and minimal polynomial

I have to find the degree of $1+\sqrt[3]{2}+\sqrt[3]{4}$ over $\mathbb{Q}$. This is what I found already: $\mathbb{Q}(1+\sqrt[3]{2}+\sqrt[3]{4})=\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{4})$ ...