# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Irr(a, K)=p(x) \in K[x]$ ...
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### 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 ...
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### 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 ...
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### 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 $[\mathbb{R}:\mathbb{Q}]=+\infty$ ??
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### 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 ...
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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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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'...
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### 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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### 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 ...
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### 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 ...
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### 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 $\aleph_1$...
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### 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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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 $K:=k(\... 1answer 54 views ### Do two isomorphic finite field extensions have the same dimension? If$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$? 1answer 47 views ### Questions on the field extension$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.$2)Find the ... 3answers 84 views ### Explain to me the difference between the notation$\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$Please explain to me the difference between the notation$\mathbb{Q}( \sqrt2) $and$ \mathbb{Q}[ \sqrt2]$. I know that these two fields are equal. But what difference do the different brackets imply? ... 1answer 123 views ### Universal property of the algebraic closure of a field At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books. Exercise 1.2a is then to convince ... 2answers 47 views ### The field closure of a countable union of countable fields is countable? If$ 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\}$is a countable family, but not a tower, ... 0answers 36 views ### element in field, not a square [duplicate] I am doing a specific exercise where the quaternion group is realised as a Galois group of some field extension. It goes like this: let$K = \mathbb Q(\sqrt 2,\sqrt 3)$and$\alpha = (2 +\sqrt 2)(3 +\...
If $F(u) \cong F(v)$, and both are subfields of a larger extension, $K$, then for $k \in K$, is $F(u)(k) \cong F(v)(k)$?
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 \$...