==Primitive root of unity==
...e element $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$.
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An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $
A root of an algebraic equation over a field $k$,
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A function that associates with an ordered pair of elements $x,y$ of the
multiplicative group $K^*$ of a
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...(Cf. also [[Algebraic number]]; [[Algebraic number theory]]; [[Extension of a field]]; [[Number field]].)
* [[Quadratic field]] — an extension of degree $n=2$;
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...oots of unity). Every element $a$ of order $n$ can be taken as a generator of this group. Then
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A field with a finite number of elements. First considered by E. Galois [[#References|[1]]].
...(p)$ with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice.
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A resolvent of an algebraic equation $ f( x) = 0 $
of degree $ n $
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''of a projective algebraic variety $X$ over a field $k$''
...References|[1]]]). Thus, this Fano variety is reducible and each component of it is not reduced at a generic point.
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...braic integer which is not a root of unity. The Mahler measure $M(\alpha)$ of an [[algebraic number]] $\alpha$ is defined by
...ds only on $f$, it is also denoted by $M(f)$ and called the Mahler measure of $f$. Jensen's formula (cf. also [[Jensen formula]]) implies the equality
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...~/encyclopedia/old_files/data/F040/F.0400280 Finite group, representation of a
A homomorphism of a finite group $ G $
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...$m$. If this congruence is not solvable, then $a$ is called a non-residue of degree $n$ modulo $m$. When $n=2$, the power residues and non-residues are
In the case of a prime modulus $p$, the question of the solvability of the congruence $x^n \equiv a \pmod p$ can be answered by using the [[Euler
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objects on the basis of their automorphism groups. For instance,
Galois theories of fields, rings, topological spaces, etc., are
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Out of 37 formulas, 36 were replaced by TEX code.-->
...number|Algebraic number]]; [[Number field|Number field]]) or a completion of such an $F$ with regard to an (infinite or finite) prime.
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Out of 90 formulas, 90 were replaced by TEX code.-->
...near space]] over a [[Field|field]] $\mathbf{F}$. Let $L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let
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...e in a suitable divisible group. If the equations stated in the definition of a divisible group have a unique solution, the group is called a $D$-group.
...polov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
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of the number $ n $,
itself. Over the field of complex numbers one has
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An [[extension of a field]] $k$ of characteristic $p \ge 0$, of the type
...of the type $\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.
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is of interest. The odd correlation of $ ( a _ {i} ) $
The goal is the design of large sets $ {\mathcal S} = \{ a ^ {( 1 ) } \dots a ^ {( m ) } \} $
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...stability of a linear closed-loop system formulated in terms of properties of the open-loop system.
...p system has $k$, $0\leq k\leq n$, roots with positive real part and $n-k$ roots with negative real part. The Nyquist criterion is as follows: The closed-lo
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A complex (sometimes, real) number that is a root of a polynomial
...s any positive integer, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.
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