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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2
votes
1answer
91 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
0
votes
1answer
41 views

Show that $p=2^k+1$

When $p$ is an odd prime and $a=Re \left ( e^{\frac{2 \pi i}{p}} \right)$ then $[\mathbb{Q}(a) : \mathbb{Q}]=\frac{p-1}{2}$. Let $\theta = \frac{2 \pi}{p}$. If $\sin{\theta}$ is a constructable ...
1
vote
1answer
30 views

Show that the equation has exactly $m$ different roots in the algebraic closure

Let $n=p^rm$, where $p$ is a prime, $m \in \mathbb{N}, r \geq 0$ an integer and $(p,m)=1$. 
I have to show that the equation $x^n=1$ has exactly $m$ different roots in the algebraic closure $\...
1
vote
1answer
69 views

Element from a formal power series that is algebraic over a field

I don't know how I should solve this exercise: The polynomial ring $K[T]$, and hence also its field of fractions $K(T)$, is a subring of $K((T))$. Give an example, for some field $K$ , of an ...
0
votes
1answer
29 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
0
votes
1answer
19 views

Separable extensions-Need help

Let $K \leq M \leq E$ be field extensions, with $K \leq E$ separable. Show that the extensions $K \leq M$ and $M \leq E$ are separable. The extension $K\leq E$ is separable if all the elements in $...
-1
votes
2answers
61 views

If $\overline{f}(x)$ is irreducible in $\mathbb{Z}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[x]$

Let $f(x)=a_0+\dots +a_n x^n \in \mathbb{Z}[x]$. Let $p$ be a prime with $p \nmid a_n$. We define $\overline{f}(x)=\overline{a_0}+\dots +\overline{a_n} x^n \in \mathbb{Z}_p[x]$ How can I show that ...
0
votes
1answer
43 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
0
votes
2answers
79 views

$\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$

I have to show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$ and then I have to find $Irr(\sqrt{5}, \mathbb{Q})$. How can I show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\...
1
vote
2answers
31 views

Why does it stand that #$\mathbb{Z}_p(a)=p^n$?

If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$ The proof is the following: Let $a \in K$. We take $\mathbb{Z}_p \...
2
votes
1answer
98 views

Real subfield of cyclotomic field is generated by $\zeta+\zeta^{-1}$

Let $p\neq 2$ a prime number, $\zeta=e^{\frac{2i\pi}{p}}$ and $\alpha=2\cos\left(\frac{2\pi}{p}\right)$. We consider the field extension $F=\mathbb Q(\zeta)$ and $E=F\cap \mathbb R$ of $\mathbb Q$. I ...
0
votes
1answer
29 views

How do we find the embeddings?

In my notes there is the following example: $$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id :...
2
votes
1answer
152 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
2
votes
1answer
66 views

Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, \sqrt{3})}]*[\...
0
votes
1answer
54 views

How to find the splitting field?

How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ?? If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following: $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$ So, the ...
5
votes
2answers
153 views

Finding a fixed subfield of $\mathbb{Q}(t)$

I'm looking at automorphisms $t \to 1-t$ and $t \to \frac{1}{t}$ of the field $\mathbb{Q}(t)$. By looking at the relations between these I think I've found the group generated by them to be $S_3$. ...
0
votes
1answer
190 views

The union of finite field extensions is a finite field extension

Assume that all elements under discussion are algebraic over $F$. Let the notation "$K=F(A)$" mean that $A\subseteq K$ and there is an injective homomorphism $\sigma:F\to K$, and every element of $K$ ...
2
votes
4answers
169 views

Roots of different irreducible polynomials are algebraically independent

Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$. Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$? Is it true ...
1
vote
0answers
32 views

Field extensions and quotient fields

STATEMENT: Suppose that $F\subseteq E$ is a field extension of $F$. And assume $u\in E$ is transcendental over $E$. Then it readily follows that $F(u)\cong F(x)$, where $F(x)$ is the quotient field of ...
1
vote
2answers
68 views

Find the splitting field of a polynomial

The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, ...
0
votes
1answer
50 views

Field of rational functions over $\mathbb{F}_p$

Let $K=\mathbb{Z}_p(x,y)$ be the field of rational functions of variables $x,y$ with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. Let $g(t)=t^p-x, h(t)=t^p-y \in K[t]$ and $E$ is the ...
0
votes
0answers
47 views

Show that the fields are equal

I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$. $$$$ To show this do I have to take first an element of $\mathbb{F}_{2^2}$...
1
vote
1answer
145 views

Basis for field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$

I'm trying to find a basis for the field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$, where $\zeta$ is the cube root of unity. I attempted this with starting with a set of elements I know ...
2
votes
2answers
362 views

Degree of field extension is infinite

If we have the field extension $\mathbb{Q}\leq \mathbb{R}$, could you explain me why it stands that $[\mathbb{R}:\mathbb{Q}]=+\infty$ ??
0
votes
1answer
36 views

Multiplicative group of a field contains maximal n-1 elements with order n

Let $F$ be a field and $n\in \mathbb N,n>1$. I want to show that the multiplicative group $K$\ $\{0\}$ contains maximal $n-1$ elements with order $n$. I actually don't have any ideas how to solve ...
0
votes
1answer
191 views

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 ...
1
vote
1answer
35 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 ...
1
vote
2answers
84 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 ...
0
votes
1answer
26 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.
1
vote
2answers
37 views

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 ...
1
vote
3answers
440 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??
2
votes
1answer
93 views

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

How can we prove 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 ...
0
votes
2answers
57 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
55 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 $[F(r):F]=4$ ...
6
votes
1answer
173 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) $x\...
2
votes
1answer
76 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 can'...
2
votes
2answers
242 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 ...
1
vote
1answer
74 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 ...
4
votes
1answer
112 views

Transcendence Degree of the Function Field of Meromorphic Functions over $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense ...
19
votes
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 $\aleph_1$...
2
votes
2answers
85 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?
0
votes
1answer
26 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 $K:=k(\...
2
votes
1answer
54 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
47 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
votes
3answers
84 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? ...
4
votes
1answer
123 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 ...
0
votes
2answers
47 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
36 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
53 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
37 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 $...