# Questions tagged [extension-field]

Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.

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### Factoring polynomials modulo 3

Let $f(x) = x^5 + 2x^2 + 2x + 2 \in\mathbb Z_3[x]$. Then the irreducible factorization of $f(x)$ is $(x^2 +1)(x^3+2x+2)$ even though it does not have a root in $\mathbb Z_3$. How did we find that ...
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Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.) I am confused by ...
1 vote
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### If a sequence is generated by a $\mathbb{Q}$-polynomial passed mod $p$, can we find an appropriate polynomial over an extension of $\mathbb{F}_{p}$?

If we have a polynomial that takes integer values for integer inputs, we can take its outputs at integer inputs and pass them $\text{mod }p$. However, my understanding is that the coefficients of the ...
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Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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### Clarification about field extension and its degree

I know there are some posts about this, but I'm still confused regarding this specific question. It is said that the dimension of any field extension $\mathbb{Q}(w)$ is the degree of the irreducible ...
1 vote
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### non isomorphic algebraically closed fields

I am trying to find all algebraically closed fields(up to isomorphism). I found that the field of all algebraic numbers over $\mathbb Q$ is algebraically closed and I also know that the field of ...
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### Fraction field and Monic Polynomial

Here is the question. $R$ is a UFD and $F$=Frac$R$, $E=F(α)$ is an algebraic extension of $F$. Prove of disprove that, if the minimal polynomial of $α$ over $F$ belongs to $R[x]$, then $α$ is a root ...
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### Vector Space and Expansion of Fields

I have learnt the complexification of real vector spaces. Let $V$ be a vector space over $\mathbb{R}$, and we can define a new vector space $V_{\mathbb{C}}$ which is a complex vector space and is ...
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### Finding subfields of an Extension Field [duplicate]

Let a $\in$ C be a root of the polynomial $X^4+ 1 \in Q[X]$ Consider the field extension Q(a) of Q. Find three fields $K_1, K_2,K_3$ such that $Q \subset K_i \subset Q(a)$ for i=1,2,3. I found out 2 ...
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### Degree of subfield fixed by single automorphism

Let $L/K$ be a finite Galois extension. If $\sigma \in \mathrm{Gal}(L/K)$ has order $d$, is it the case that $$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\}$$ satisfies $[L^\sigma:K] = d$? This ...
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### Given $K/F$ where $K$ is a field extension of $F$ of the form $F[x]/\langle p(x) \rangle$, what is the structure of $K[x]$?

$F$ is a field. $\langle p(x) \rangle$ is a maximal ideal. So $K = F[x]/\langle p(x) \rangle$ is a field extension. I am trying to understand what would be the structure of $K[x]$? $F[x]$ has all ...
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### What are the $\Bbb Q$-automorphisms of $\Bbb Q(\alpha)$ with $\alpha$ a root of $x^3-3x^2+3$?

I know the polynomial is irreducible over $\Bbb Q$ and the roots are real irrational. A $\Bbb Q$-automorphism $\sigma:\Bbb Q(\alpha)\to \Bbb Q(\alpha)$ is determined by $\sigma(\alpha)$, which is also ...
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### Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?

Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
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1 vote
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### Are field isomorphisms necessarily linear isomorphisms?

Let’s say we have a finite field extension $L/K$. Suppose we have that $L(\alpha )$ and $L(\beta )$ are isomorphic as fields (finite extensions) with an isomorphism $\phi$. Now viewing $L(\alpha )$ ...
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### Every number field of degree $3$ is of the form $\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $x^3 + ax + b$

I want to show that every number field of degree $3$ is of the form $\mathbb{Q}(\alpha)$ where the minimal polynomial of $\alpha$ is of the form $m_\alpha(x) = x^3 + ax + b$. Let $K$ be the cubic ...
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### differential extensions meaning, use, validity of some expressions, derivation, := meaning

In the attached image IK means field characteristic zero, [...] means polynomial and IK[x,y] means polynomial in x and y in IK and $m(x,y)$ is generally set to $0$. If you need more detail or to see ...
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### If $F$ is algebraic over $K$ and $D$ is an intermediate and an integral domain, is $D$ a field?

I would imagine it would be as simple as showing that $D$ is finite since a finite integral domain is a field. I know that if $F$ is a finite extension of $K$, then $F$ is an algebraic extension of $K$...
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### Find a particular intermediate field $M$ such that $\mathbb{Q}\subset M\subset\mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})$.

Problem: Let $\alpha=\sqrt{\frac{3+i\sqrt{7}}{2}}$ and $K=\mathbb{Q}(\alpha)$. Find the fixed field $M=\{x\in \mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})|\sigma(x)=x\}$, where $\sigma$ is the ...
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### Question on the irreducibility of $x^4 - 10x^2 +1$

I am currently doing some work in Galois theory, and the following situation has me perplexed. The polynomial $~x^4 -10x^2 + 1$ is irreducible over $\mathbb Q,$ which can be shown in a variety of ...
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### L is normal extension if it's splitting field

I am trying to understand this proof That if L is splitting field of some polynomial then it's normal extension . I got the part till we get that $L(\alpha)/K(\alpha)$ and $L(\beta )/K(\beta)$ are ...
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### Possible $|F(\alpha, \beta):F|$ where $|F(\alpha):F|=6$ and $|F(\beta):F|=15$?

Here is a past paper problem which I am struggling to solve currently. Let $\alpha,\beta\in E$, where $E$ an extension of field $F$. We are given $|F(\alpha):F|=6$ and $|F(\beta):F|=15$. What are the ...
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I am a completing a past paper question and I am undecided on what method to use here. The question is: For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are: (1) Check each $a\... 1 vote 1 answer 57 views ### Is$\mathbb{Q}(\sqrt{3},\sqrt{3},i)$a splitting field over$\mathbb{Q}?$Is$\mathbb{Q}(\sqrt{3},\sqrt{3},i)$a splitting field over$\mathbb{Q}$? I am thinking that if i prove$\mathbb{Q}(\sqrt{3})$is a splitting field over$\mathbb{Q}$and$\mathbb{Q}(\sqrt{3},i)...
I would like to extend the question here. Assume that $X\rightarrow Spec(K)$ is proper and X is reduced. Since is proper and $Spec(K)$ is Noetherian, $X$ is Noetherian and hence it will be finite ...
### Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. [duplicate]
Problem: Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. My Attempt: The roots for $f(t)=t^6-2t^3-1\; \text{in}\; \mathbb{C}$ are \$\{\alpha, \beta, \zeta\alpha, \zeta\beta, \zeta^2\...