I was trying to read the question below If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$ and got stuck at the notation $|F|$. I know if it's used in real or complex number it just means the magnitude of the number but I'm confused when it's used in the context of a field $F$.
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1It refers to the number of elements in $F$. – lulu Jan 24 '24 at 13:24
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It's the usual notation for the cardinality of a set (number of elements).
Vladimir Lysikov
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So for example if F is a set of integers ranging from 0 to 1000 does it mean |F| = 1001? – Helloworld Jan 24 '24 at 13:57
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Yes. It is also not a field, but this is not relevant to this specific notation. – Vladimir Lysikov Jan 24 '24 at 14:04
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Wait I seem to have some misunderstanding of the concept of field can field F be described as set of F^1, F^2, F^3,… and so on? – Helloworld Jan 24 '24 at 15:21
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Field is a set with two operations (addition and multiplication) which satisfy the usual addition and multiplication laws. For example, $\mathbb{Q}$ or $\mathbb{R}$ are fields. There are also finite fields, the simplest examples are the sets of residues modulo a prime number $p$ (for example in a field with 2 elements we have $0 + 0 = 1 + 1 = 0$, $0 + 1 = 1 + 0 = 1$, $0\cdot 0 = 0\cdot 1 = 1\cdot 0 = 0$, $1\cdot 1 = 1$). If a field $F$ is finite, then it contains a special element $g$, called primitive element, such that $F = {0, 1, g, g^2, g^3, \dots, g^{|F| - 2}}$. – Vladimir Lysikov Jan 24 '24 at 16:55
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It's better if you write a new question if you have question about fields. – Vladimir Lysikov Jan 24 '24 at 16:56