# Tagged Questions

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### What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
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### Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
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### Show the following subspaces are invariant

Let $V$ be a vector space over a field $F$ and let $\alpha \in End(V)$. IF $W$ and $Y$ are subspaces of $V$ which are invariant under $\alpha$, show that both $W+Y$ and $W\cap Y$ are invariant under ...
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### Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
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### What kind of field is denoted by $\mathbb{Z}_2^n$?

For a homework question I have to prove whether or not some set is a vector space over a field $\mathbb{Z}_2^n$, but I am not sure what this notation means and the textbook doesn't seem to clarify. ...
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### Uniqueness of complements in general vector spaces

Motivation I asked the question: Is my proof correct: rank-nullity in a field $K$. Although the answer given by Marc van Leeuwen made perfect sense, it made me wonder about one thing: Introduction ...
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### Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself? Intuitively, it seems obvious because the definition of a field is nearly the same as that of a vector space, just with scalers instead of ...
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### Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
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### Field structure of vectors in $\mathbb{R}^3$

Probably a trivial question: By representing vectors in $\mathbb{R}^2$ as complex numbers we can define multiplication of vectors so that $\mathbb{R}^2$ has a field structure. Can this be extended to ...
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### Injection from an integral domain to its field of fractions.

I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
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### Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ with the following property?

Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ (proper subset) as follows: $\mathbb{R}$ is a vector space over $K$ and has no finite generating set and $K$ is a vector ...
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### Linearly dependent vectors over finite fields

My problem is as follows: Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$. Every vector in this vector space can be regarded as an element of the extension field ...
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### Basis for $\mathbb{Q}(\sqrt{2},\sqrt{3})$

How can I find a basis for the field extension $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$? I can show that [$\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=4$ (so I am looking for 4 basis elements). ...
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### Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...