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Apparently, all fields are vector spaces over themselves.

But the definition of a vector space involves the operation of vector addition and scalar multiplication, and elements of a vector space are considered to be vectors.

So, when we say that all fields are vector spaces over themselves, are we in that instant considering the elements of the field to be vectors (where we might not have been when we were considering the field in relation to a separate vector space)? I get that once you just take vector addition to be regular field addition, scalar multiplication to be field multiplication and field elements to be vectors the axioms are satisfied, but no one seems to mention this move.

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    $\begingroup$ You are correct: "once you just take vector addition to be regular field addition, scalar multiplication to be field multiplication and field elements to be vectors the axioms are satisfied". $\endgroup$
    – GEdgar
    Jul 29, 2022 at 10:30
  • $\begingroup$ OK, but the elements of R, for example, are numbers, right? So why are we now taking them to be vectors without comment? $\endgroup$
    – Dan Öz
    Jul 29, 2022 at 10:51
  • $\begingroup$ @DanÖz They're numbers and they're vectors. They're not mutually exclusive concepts. $\endgroup$
    – CyclotomicField
    Jul 29, 2022 at 10:54
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    $\begingroup$ @Mr.GandalfSauron Your construction forces $b=0$ so it can't be arbitrary. This is unsurprising since vector spaces require a zero vector so either $a$ or $b$ had to fill that role. $\endgroup$
    – CyclotomicField
    Jul 29, 2022 at 11:55
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    $\begingroup$ @CyclotomicField By "anything" I meant that it is just a symbol. Of course $b$ is the "zero" element. I meant that instead of writing it as "b" or $0$ we can write it as any symbol we want(say a smile face) . So even though they are not necessarily "arrows with magnitudes" , they are vectors as they satisfy the axioms. $\endgroup$
    – Mr.Gandalf Sauron
    Jul 29, 2022 at 12:10

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