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| In [[mathematics]], a '''Witt vector''' is an [[infinite sequence]] of elements of a [[commutative ring]]. [[Ernst Witt]] showed how to put a ring [[mathematical structure|structure]] on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order ''p'' is the ring of [[p-adic integer|''p''-adic integers]].
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| ==Motivation==
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| We basically want to derive the ring <math>p</math>-adic integers <math>\mathbb{Z}_p</math> from the finite field with <math>p</math> elements, <math>\mathbb{F}_p</math>, by some '''general construction'''.
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| A <math>p</math>-adic integer is a sequence <math>(n_0,n_1,...)</math> with <math>n_i\in\mathbb{Z}/p^{(n+1)}\mathbb{Z}</math>,such that <math>n_i\equiv n_j\mod p^i</math> if <math>i<j</math>. They can be expanded as a [[power series]] in <math>p</math>:
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| <math>a_0 + a_1 p^1 + a_2 p^2 + ...</math>, where the <math>a</math>'s are usually taken from the set <math>\{0, 1, 2, ..., p-1\}</math> (The equation is happening in <math>\mathbb{Z}_p</math>, with <math>a_i</math> and <math>p^j</math> all images from <math>\mathbb{Z}</math> to <math>\mathbb{Z}_p</math>). '''Set-theoretically''' it is <math>\mathbb{F}_p</math>. But <math>\mathbb{Z}_p</math> is not isomorphic to <math>\prod_{\mathbb{N}}\mathbb{F}_p</math>. If we denote <math>a+b=c</math>, then the addition should instead be:
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| :<math>
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| c_0 \equiv a_0+b_0 \mod p
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| </math>
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| :<math>
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| c_0+c_1 p\equiv a_0+a_1 p+b_0+b_1 p \mod p^2
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| </math>
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| :<math>
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| c_0+c_1 p+c_2 p^2 \equiv a_0+a_1 p+a_2 p^2+b_0+b_1 p+b_2 p^2 \mod p^3
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| </math>
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| But we lack some properties of the coefficients to produce a general formula.
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| Luckily, there is an alternative subset of <math>\mathbb{Z}_p</math> which can work as the coefficient set. This is the [[Teichmüller character|Teichmüller representatives]] of elements of <math>\mathbb{F}_p</math>. Without <math>0</math> they form a subgroup of <math>\mathbb{Z}_p^*</math>, identified with <math>\mathbb{F}_p^*</math> through the [[Teichmüller character]] <math>\omega:\mathbb{F}_p^*\rightarrow\mathbb{Z}_p^*</math>. Note this is '''not''' additive, as the sum need not be a representative. Despite this, if <math>\omega(k)=\omega(i)+\omega(j)\mod p</math> in <math>\mathbb{Z}_p</math>, then <math>i+j=k</math> in <math>\mathbb{F}_p</math>. This is conceptually justified by <math>m\circ \omega=\mathrm{id}_{\mathbb{F}_p}</math> if we denote <math>m:\mathbb{Z}_p\rightarrow\mathbb{Z}_p/p\mathbb{Z}_p\cong\mathbb{F}_p</math>.
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| They are explicitly calculated as roots of <math>x^{p-1}-1=0</math> through [[Hensel's lemma#Hensel Lifting|Hensel lifting]]. For example, in <math>\mathbb{Z}_3</math>, to calculate the representative of <math>2</math>, you first find the unique solution of <math>x^{2}-1=0</math> in <math>\mathbb{Z}/9\mathbb{Z}</math> with <math>x\equiv 2\mod 3</math>; You get <math>8</math>, then repeat it in <math>\mathbb{Z}/27\mathbb{Z}</math>, with conditions <math>x^{2}-1=0</math> and <math>x\equiv 2\mod 9</math>; This time it is <math>26</math>, and so on. The existence of lift in each step is guaranteed by <math>(x^{p-1}-1,(p-1)x^{p-2})=1</math> in every <math>\mathbb{Z}/p^n\mathbb{Z}</math>.
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| We can also write the representatives as <math>a_{0} + a_{1} p^1 + a_{2} p^2 + ...</math>. Note for every <math>j\in\{0, 1, 2, ..., p-1\}</math>, there is exactly one representative, namely <math>\omega(j)</math>, with <math>a_{0}=j</math>, so we can also expand every <math>p</math>-adic integer as a power series in <math>p</math>, with coefficients from the Teichmüller representatives.
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| Explicitly, if <math>b=a_{0} + a_{1} p^1 + a_{2} p^2 + ...</math>, then <math>b-\omega(a_0)=a'_{1} p^1 + a'_{2} p^2 + ...</math>. Then you subtract <math>\omega(a'_1)p</math> and proceed similarly. Note the coefficients you get most probably differ from the <math>a_i</math>'s, except the first one.
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| This time we have additional properties of the coefficients like <math>a_i^p=a_i</math>, so we can make some changes to get a neat formula. Since the Teichmüller character is '''not''' additive, we don't have <math>c_0=a_0+b_0</math> in <math>\mathbb{Z}_p</math>. But it happens in <math>\mathbb{F}_p</math>, as the first congruence implies. So actually <math>c_0^p\equiv (a_0+b_0)^p \mod p^2</math>, thus <math>c_0-a_0-b_0\equiv (a_0+b_0)^p-a_0-b_0\equiv \binom{p}{1} a_0^{p-1}b_0+...+ \binom{p}{1} a_0 b_0^{p-1} \mod p^2</math>. Since <math>\binom{p}{i}</math> is divisible by <math>p</math>, this resolves the <math>p</math>-coefficient problem of <math>c_1</math> and gives <math>c_1\equiv a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1}\mod p</math>. Note this completely determines <math>c_1</math> by the lift. Moreover, the <math>\mod p</math> indicates that the calculation can actually be done in <math>\mathbb{F}_p</math>, satisfying our basic aim.
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| Now for <math>c_2</math>. It is already very cumbersome at this step. <math>c_1=c_1^p \equiv (a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1})^p\mod p</math>. As for <math>c_0</math>, a single <math>p</math>th power is not enough: actually we take <math>c_0=c_0^{p^2}\equiv(a_0+b_0)^{p^2}</math>. <math>\binom{p^2}{i}</math> is not always divisible by <math>p^2</math>, but that only happens when <math>i=pd</math>, in which case <math>a^ib^{p^2-i}=a^db^{p-d}</math> combined with similar monomials in <math>c_1^p</math> would make a multiple of <math>p^2</math>.
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| At this step, we see that we are actually working with something like
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| :<math>
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| c_0 \equiv a_0+b_0 \mod p
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| </math>
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| :<math>
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| c_0^p+c_1 p\equiv a_0^p+a_1 p+b_0^p+b_1 p \mod p^2
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| </math>
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| :<math>
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| c_0^{p^2}+c_1^p p+c_2 p^2 \equiv a_0^{p^2}+a_1^p p+a_2 p^2+b_0^{p^2}+b_1^p p+b_2 p^2 \mod p^3
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| </math>
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| This motivates the definition of Witt vectors.
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| ==Construction of Witt rings==
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| Fix a [[prime number]] ''p''. A '''Witt vector''' over a commutative ring ''R'' is a sequence :<math> (X_0,X_1,X_2,...)</math> of elements of ''R''. Define the '''Witt polynomials''' <math>W_i</math> by
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| #<math> W_0=X_0\,</math>
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| #<math> W_1=X_0^p+pX_1</math>
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| #<math> W_2=X_0^{p^2}+pX_1^p+p^2X_2</math>
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| and in general
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| :<math> W_n=\sum_ip^iX_i^{p^{n-i}}.</math>
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| <math> (W_0,W_1,W_2,...)</math> is called the '''ghost components''' of the Witt vector <math>(X_0,X_1,X_2,...)</math>, and is usually denoted by <math> (X^{(0)},X^{(1)},X^{(2)},...)</math>.
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| Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring ''R'' into a ring, called the '''ring of Witt vectors''', such that
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| *the sum and product are given by polynomials with integral coefficients that do not depend on ''R'', and
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| *Every Witt polynomial is a homomorphism from the ring of Witt vectors over ''R'' to ''R''.
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| In other words, if
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| *<math> (X+Y)_i</math> and <math> (XY)_i</math> are given by polynomials with integral coefficients that do not depend on ''R'', and
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| *<math> X^{(i)}+Y^{(i)}=(X+Y)^{(i)}</math>, <math> X^{(i)}Y^{(i)}=(XY)^{(i)}</math>.
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| The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
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| *<math>(X_0,X_1,...)+(Y_0,Y_1,...)=(X_0+Y_0,X_1+Y_1+(X_0^p+Y_0^p-(X_0+Y_0)^p)/p,...)</math>
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| *<math>(X_0,X_1,...)\times(Y_0,Y_1,...)=(X_0 Y_0,X_0^p Y_1+X_1 Y_0^p+p X_1 Y_1,...)</math>.
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| ==Examples==
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| *The Witt ring of any commutative ring ''R'' in which ''p'' is invertible is just isomorphic to ''R''<sup>'''N'''</sup> (the product of a countable number of copies of ''R''). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to ''R''<sup>'''N'''</sup>, and if ''p'' is invertible this homomorphism is an isomorphism.
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| *The Witt ring of the [[finite field]] of order ''p'' is the ring of ''p''-adic integers, as is demonstrated above.
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| *The Witt ring of a finite field of order ''p''<sup>''n''</sup> is the [[Splitting of prime ideals in Galois extensions|unramified extension]] of degree ''n'' of the ring of ''p''-adic integers.
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| ==Universal Witt vectors==
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| The Witt polynomials for different primes ''p'' are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime ''p'').
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| Define the universal Witt polynomials ''W''<sub>''n''</sub> for ''n''≥1 by
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| #<math> W_1=X_1\,</math>
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| #<math> W_2=X_1^2+2X_2</math>
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| #<math> W_3=X_1^3+3X_3</math>
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| #<math> W_4=X_1^{4}+2X_2^2+4X_4</math>
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| and in general
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| :<math> W_n=\sum_{d|n}dX_d^{n/d}.</math>
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| Again, <math>(W_1,W_2,W_3,...)</math> is called the '''ghost components''' of the Witt vector <math>(X_1,X_2,X_3,...)</math>, and is usually denoted by <math> (X^{(1)},X^{(2)},X^{(3)},...)</math>.
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| We can use these polynomials to define the ring of universal Witt vectors over any commutative ring ''R'' in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring ''R'').
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| ==Generating Functions==
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| Later Witt orally stated another approach using generating functions.<ref>{{cite book |last=Lang |first=Serge |title=Algebra |publisher=Springer|edition = 3rd |date=September 19, 2005 |pages=330 |chapter=Chapter VI: Galois Theory |isbn=978-0-387-95385-4}}</ref>
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| ===Definition===
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| Let <math>X</math> be a Witt vector and define
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| :<math>f_X(t)=\prod_{n\ge 1}(1-X_n t^n)=\sum_{n\ge 0}A_n t^n</math> | |
| For <math>n\ge 1</math> let <math>\mathcal{S}_n</math> denote the collection of subsets of <math>\{1,2,...,n\}</math> whose elements add up to <math>n</math>. Then <math>A_n=\sum_{S\in\mathcal{S}}(-1)^{|S|}\sum_{i\in S}{X_i}</math>.
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| We can get the ghost components by taking the [[logarithmic derivative]]:
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| :<math>\frac{d}{dt}\log f_X(t)=\sum_{n\ge 1}\frac{d}{dt}(1-X_n t^n)=-\sum_{n\ge 1}\sum_{d\ge 1}\frac{X_n^d t^{nd}}{d}=-\sum_{m\ge 1}\frac{\sum_{d|m}\frac{m}{d}X_{\frac{m}{d}}^d}{m}t^m=-\sum_{m\ge 1}\frac{X^{(m)}t^m}{m}</math>
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| ===Sum===
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| Now we can see <math>f_{Z}(t)=f_X(t) f_Y(t)</math> if <math>Z=X+Y</math>. So that <math>C_n=\sum_{0\le i\le n}A_n B_{n-i}</math> if <math>A_n,B_n,C_n</math> are respective coefficients in the power series for <math>f_X(t),f_Y(t),f_Z(t)</math>. Then <math>Z_n=\sum_{0\le i\le n}A_n B_{n-i}-\sum_{S\in\mathcal{S},S\ne\{n\}}(-1)^{|S|}\sum_{i\in S}{Z_i}</math>. Since <math>A_n</math> is a polynomial in <math>X_1,...,X_n</math> and likely for <math>B_n</math>, we can show by induction that <math>Z_n</math> is a polynomial in <math>X_1,...,X_n,Y_1,...,Y_n</math>.
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| ===Product===
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| If we set <math>W=XY</math> then
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| :<math>\frac{d}{dt}\log f_W(t)=-\sum_{m\ge 1}\frac{X^{(m)}Y^{(m)}t^m}{m}</math>
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| But
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| :<math>\sum_{m\ge 1}\frac{X^{(m)}Y^{(m)}}{m}t^m=\sum_{m\ge 1}\frac{\sum_{d|m}d X_d^{m/d}\sum_{e|m}e Y_e^{m/e}}{m}t^m</math>
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| Now 3-tuples <math>{m,d,e}</math> with <math>m\in\mathbb{Z}^+,d|m,e|m</math> are in bijection with 3-tuples <math>{d,e,n}</math> with <math>d,e,n\in\mathbb{Z}^+</math>, via <math>n=m/[d,e]</math> (<math>[d,e]</math> is the [[Least common multiple]]), our series becomes
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| :<math>\sum_{d,e\ge 1}\frac{\frac{d e}{ [d,e]}\sum_{n\ge 1} (X_d^{ [d,e]/d } Y_e^{ [d,e]/e } t^{ [d,e] })^n}{n}</math>
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| So that
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| :<math>f_W(t)=\prod_{d,e\ge 1}(1-X_d^{[d,e]/d}Y_e^{[d,e]/e} t^{[d,e]})^{d e/[d,e]}=\sum_{n\ge 0}D_n t^n</math>
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| where <math>D_n</math>s are polynomials of <math>X_1,...,X_n,Y_1,...,Y_n</math>. So by similar induction, suppose <math>f_W(t)=\prod_{n\ge 1}(1-W_n t^n)</math>, then <math>W_n</math> can be solved as polynomials of <math>X_1,...,X_n,Y_1,...,Y_n</math>.
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| ==Ring schemes==
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| The map taking a commutative ring ''R'' to the ring of Witt vectors over ''R'' (for a fixed prime ''p'') is a [[functor]] from commutative rings to commutative rings, and is also representable, so it can be thought of as a [[ring scheme]], called the '''Witt scheme''', over Spec('''Z'''). The Witt scheme can be canonically identified with the spectrum of the [[ring of symmetric functions]].
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| Similarly the rings of truncated Witt vectors, and the rings of
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| universal Witt vectors, correspond to ring schemes, called the '''truncated Witt schemes''' and the '''universal Witt scheme''' .
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| Moreover, the functor taking the commutative ring <math>R</math> to the set <math>R^n</math> is represented by the affine space <math>\mathbb{A}_{\mathbb{Z}}^n</math>, and the ring structure on ''R<sup>n</sup>'' makes <math>\mathbb{A}_{\mathbb{Z}}^n</math> into a ring scheme denoted <math>\underline{\mathcal{O}}^n</math>. From the construction of truncated Witt vectors it follows that their associated ring scheme <math>\mathbb{W}_n</math> is the scheme <math>\mathbb{A}_{\mathbb{Z}}^n</math> with the unique ring structure such that the morphism <math>\mathbb{W}_n\rightarrow \underline{\mathcal{O}}^n</math> given by the Witt polynomials is a morphism of ring schemes.
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| ==Commutative unipotent algebraic groups==
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| Over an [[algebraically closed field]] of characteristic 0, any [[unipotent]] abelian connected [[algebraic group]] is isomorphic to a product of copies of the additive group <math>G_a</math>.
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| The analogue of this for fields of characteristic ''p'' is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However these are essentially the only counterexamples: over an algebraically closed field of characteristic ''p'', any [[unipotent]] abelian connected [[algebraic group]] is
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| [[Isogeny#Abelian varieties up to isogeny|isogenous]] to a product of truncated Witt group schemes.
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| ==See also==
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| *[[Formal group]]
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| *[[Artin–Hasse exponential]]
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| ==References==
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| {{reflist}}
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| *{{springer|authorlink=Dolgachev|first=I.V. |last=Dolgachev|id=Witt_vector|title=Witt vector}}
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| *{{citation|mr=2553661
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| |last=Hazewinkel|first= Michiel
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| |chapter=Witt vectors. I.|title= Handbook of algebra. Vol. 6|pages=319–472,
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| |publisher= Elsevier/North-Holland|place= Amsterdam|year= 2009|arxiv=0804.3888|isbn=978-0-444-53257-2}}
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| * {{Citation | last1=Mumford | first1=David | author1-link = David Mumford | title=Lectures on Curves on an Algebraic Surface | publisher=[[Princeton University Press]] | location=Princeton, NJ | series=Annals of Mathematics Studies | isbn=978-0-691-07993-6 | volume=59 | year=}}
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| * {{Citation | last1=Serre | first1=Jean-Pierre | author1-link = Jean-Pierre Serre | title=Local fields | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90424-5 | mr=554237 | year=1979 | volume=67}}, section II.6
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| * {{Citation | last1=Serre | first1=Jean-Pierre | title=Algebraic groups and class fields | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-96648-9 | mr=918564 | year=1988 | volume=117}}
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| * {{Citation | url=http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725 | last1=Witt | first1=Ernst | author1-link = Ernst Witt | title=Zyklische Körper und Algebren der Characteristik p vom Grad p<sup>n</sup>. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p<sup>n</sup> | language=German | year=1936 | journal=Journal für Reine und Angewandte Mathematik | volume=176 | pages=126–140}}
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| *Greenberg, M. J. (1969), ''Lectures on Forms in Many Variables'', New York and Amsterdam, Benjamin, {{MathSciNet|id=241358}}, ASIN: B0006BX17M
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| [[Category:Ring theory]]
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| [[Category:Algebraic groups]]
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| [[Category:Combinatorics on words]]
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