# 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 ...

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### Irreducible or reducible polynomials in $\mathbb Z_3[X]$ of degree 0,1,2,3

In $\mathbb Z_3[x]$, find all polynomials and classify reducible or irreducible for all polynomials of degree less than 4. Here is what I am thinking Def of Irreducible Let F be a field. A ...
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### Extensions of $\mathbb{Q}((T))$ and $\mathbb{F}_p((T))$

Okay, I'm having some trouble finding good references for this, so here goes: Is every finite extension of $\mathbb{Q}((T))$ isomorphic to $K((T^{1/e}))$ where $K$ is finite over $\mathbb{Q}$, and ...
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### Ordered field, Bounded set, and the containment

I am now in engineering mathematics class and it goes over some basic set theory. Since I haven't had any experience with set theory, three statements leave me confused. Thanks for your help! ...
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### Trace of an element of a number field is in $\mathbb{Q}$. A strange proof

I found a proof (Here, at the beginning) (it's about the discriminant but still applies for the trace) of the well-known fact that $\text{Tr}_{K|\mathbb{Q}}(\alpha) \in \mathbb{Q}$ which seems to use ...
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### Subfields of **C** which extend to **C** via finite extensions

The field Q is a subfield of C but it is in a sense "much smaller" than C. The field R however has a finite extension of order just two to the field C. My question is: are there other subfields of C ...
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### How does the determinant change with respect to a base change?

Problem Suppose $k$ is a (commutative) field, and $A$ is a finite (dimensional) commutative unitary $k$-algebra. $M=A^n$ is a free $A$-module, and therefore can be seen as a finite-dimensional ...
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### Why is the field norm well-defined?

Studying Lang's algebra at page 285, I have a question about the field norm. The statement I wonder is as follows: Let E be a finite extension of k. Then $N_k^E : E^* \rightarrow k^*$ is a ...
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### Out of this list of four fields, which two are equal: $\Bbb Q[ \sqrt{20}]$, $\Bbb Q[\sqrt{10}]$, $\Bbb Q[ \sqrt2,\sqrt5]$, $\Bbb Q[ \sqrt2+\sqrt5]$?

What I have so far is: $$\Bbb{Q}[\sqrt{20}] = \Bbb{Q}[ √ 5] = \Bbb{Q}[ \sqrt4\cdot\sqrt5] = \Bbb{Q}[ 2\cdot\sqrt5].$$ $$\Bbb{Q}[\sqrt2+ \sqrt5] \implies (\sqrt2+ \sqrt5)^2 = 7 + 2\sqrt{10}$$ so ...
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### Show that any finite purely inseparable extensions has a $p-$basis.

A set $\{a_1,...,a_n\} \subseteq K$ is said to be a $p-$basis for $K/F$ provided that there is a chain of proper extensions $F \subset F(a_1) \subset \cdots \subset F(a_n)=K$. Show that any finite ...
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### What is the minimal polynomial of $\alpha = \frac{3^{1/2}}{1+2^{1/3}}$ over $\mathbb{Q}$?

I've tried to solve this by algebraic manipulation: putting the relation in equation form, raising it to powers, rearranging terms, rewriting some of them in terms of $\alpha$ and reading off the ...
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### Finding the number of a roots in an equation modulo n

I am looking for some n such that $$x^2+x=0\pmod{n}$$ has at least 4 solutions. Is there any way of doing this reasonably quickly without having to check every solution manually? There must be. ...
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### Without determinants, show a set in $L^n \subset K^n$ is linearly independent over $K$ if it is over $L$.

Let $L$ and $K$ be fields with $L \subset K$. Let $v_1,\ldots,v_r \in L^n$ be column vectors, linearly independent over $L$. Of course, we can also consider the vectors to sit in $K^n \supset L^n$. ...
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### Describe the subfields of $\mathbb{C}$ of the form: $\mathbb{Q}(\alpha)$ where $\alpha$ is the real cube root of $2$.

Describe the subfields of $\mathbb{C}$ of the form: $\mathbb{Q}(\alpha)$ where $\alpha$ is the real cube root of $2$. Let $\alpha$ be the real cube root of $2$, and consider $\mathbb{Q}(\alpha)$. As ...
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### What does this field notation mean?

I am reading a problem and its solution posted online here that says: Problem 3: Give an example of a Noetherian ring R that contains a subring that is not Noetherian. And then, Solution: ...
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### Number of invertible matrices in a field ( char is not 2) which some of every two is not invertible.

Why is there at most C(2n,n) n*n matrices in a field (char is not 2) in which sum of every two is not invertible?
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### Correspondence between Polynomial Roots and Automorphism Group

Suppose you have a field $F$, and a value $\eta$ such that $f$ is a polynomial of minimal degree, with coefficients in $F$ such that $f(\eta)=0$. Then it is claimed that for every unique root $q$ of ...
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### Is $GL(\mathbb R)$ a non-commutatif field?

Is $GL(\mathbb R)$ a non-commutatif field ? Since $GL(\mathbb R)$ is a ring an that all element are invertible, I would says that it's a field, but since elements doesn't commute for $\cdot$, I would ...
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### Let $E$ be an algebraic extension of $F$. If every polynomial in $F[x]$ splits in $E$, show that $E$ is algebraically closed.

Let $E$ be an algebraic extension of $F$. If every polynomial in $F[x]$ splits in $E$, show that $E$ is algebraically closed. This question appear in the Gallian's contemporary abstract algebra. ...
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### $L/K$ is a field extension. What does a $K$-automorphism of $L$ look like?

If $L/K$ is a field extension, then $L$ can be considered as a $K$-vectorspace of dimension $[L:K]$. If we consider $K$-automorphisms of $L$, they take $\overline\sigma: L\to L$ where ...
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### Comparing degrees of field extensions: compositum over field versus field over intersection.

I'm stuck on the following question: For fields $L$ and $K$, show that it is possible that $[LK:K]<[L:L\cap K]$. I have realized that it is not possible if $L$ is an extension of $K$. This ...
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### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}$, special cases.

What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}$$in the following two cases? When $ab \neq 0$ and $p = 2$. When $ab = 0$.
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### Definition of kernels of ring and field homomorphisms

Let $\varphi:A\to B$ a ring morphism. Why do we define $$\ker\varphi=\{x\in A\mid \varphi(x)=0\}$$ and not $$\ker\varphi=\{x\in A\mid \varphi(x)=1\} ?$$ Maybe it's a consequence of the fact ...
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### What is the minimal polynomial of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$? [on hold]

What is the minimal polynomial over $\mathbb{Q}$ of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$, where $\zeta_j$ is a $j$-th primitive root of unity for each $j$? I want to say it should be ...
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### How to determine if two (irreducible) polynomials give rise to the same (field) ring

Write down all polynomials of exactly degree $3$ over $\Bbb F_2$. How many different rings $\Bbb F_2[x]/\langle f\rangle$ can we form up to isomorphism, and which of these are fields. \begin{align*} ...
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### Finding an inverse of a polynomial modulo some other polynomial

1) $x^3-3x-1$ is irreducible by the rational root test. 2) $L=\Bbb Q[x]/\langle x^3-3x-1\rangle$ is therefore a field. 3) $f(x) = x^4+2x^3+3\in L$. $f(x)\cong 3x^2+7x+5 + \langle x^3-3x-1\rangle$ ...
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### Unique factorization domains - Definition understanding

The point of a UFD is that any element can be rewritten as a product of irreducible factors, where any other product of irreducible factors is just a rearrangement of the exact same terms, is this ...
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### What is the field's role in a linear space?

A linear space $(U,F,\oplus,\odot)$ is a set of vectors $U$ and a field $(F,+,\cdot)$ for which vector addition $\oplus:V\times V\to V$ and scalar multiplication $\odot:F\times V\to V$ are defined. ...
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### Show that it could be that $[LK:K] \lt [L: L\cap K]$

How could it be that $[LK:K] \lt [L: L\cap K]$? A simple Venn diagram (indicating overlap of the fields) of an $L$ and a $K$ field seems to indicate that the part of $L$ which isn't in $L\cap K$ is ...
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### $f(x)$ has factor $x-a$ iff $R$ is UFD?

Let $R$ be a commutative ring with identity. What is the condition on $R$ so that the following statement is valid? Statement: Let $f(x) \in R[x]$. Then $f(x)$ has factor $x-a$ if and only if ...
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### To find an Ideal which is the Kernel of Homomorphism that defines Isomorphism

$R=\{a+\sqrt2 b|a,b\in \mathbb{R}\}$ is an integral-domain. I want to find and ideal $I=<f(x)>| f(x)\in Q[x]$ so $$Q[x]/I\tilde{=}R$$ The quotient-group isomorphic to $R$. So I started from ...
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### Is $x^6-3$ irreducible over $\mathbb{F}_7$? [duplicate]

I know that $\mathbb{F}_7=\mathbb{Z}_7$, and the all possible solutions of $x^6-1=0$ over $\mathbb{Z}_7$ are 1~6, so if we let the root of equation $x^6-3$ as $t$ then the solutions of $x^6-3=0$ is ...
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### Applications for $\mathbb Q(\sqrt 2)$

While it's certainly interesting that we can extend fields in surprising ways, are the "first example" type field extensions actually useful for anything? In particular, what about the field \Bbb ...
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### If $K$ is a finite field, then $|K|=p^d$ where $p$ is prime and $d\geq 1$. [duplicate]

Let $K$ a finite field. I want to show that $|K|=p^d$. I consider an homomorphism $\Phi:\mathbb Z\to K$ which is clearly not injective, therefore $\ker\Phi\neq\{0\}$. What I want is to prove that ...
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### If $x$ is algebraic over $K$, show that $K[x]$ is a field.

Let $L/K$ an extension field (and $K$ a domain). $x$ is algebraic over $K$ if there is a polynomial $P(X)\in K[X]\backslash \{0\}$ such that $P(x)=0$. I want to show that $K[x]$ is a field. I know ...