# Element Constructions and Conversions Local ring elements are implemented using a balanced mod representation. This allows small negative elements $x$ to be represented as $x$ rather than $p^k - x$ where $k$ is a $p$-adic precision. ## Constructions To simplify the creation of elements in a local ring, various coercions are provided. The most obvious is to regard the integer ring or rational field as embedded in the $p$-adic ring or field. But there is a range of coercions available, including that of elements from the residue class field. To create an element of a local field not lying in the local ring, constructors are provided that create an element coercible into the ring, and which increase or decrease this element’s valuation in the field. ### `Zero(L): RngPad -> RngPadElt` ### `Zero(L): RngPadRes -> RngPadResElt` ### `Zero(L): RngPadResExt -> RngPadResExtElt` ### `Zero(L): FldPad -> FldPadElt` Given a local ring or field $L$, create the additive identity of $L$. Note that if $L$ is an unbounded precision ring, this will only be the zero element to the default precision of the ring, and hence only an approximation to the additive identity of $L$. The valuation and hence the absolute precision of the zero of a field is infinite. ### `One(L): RngPad -> RngPadElt` ### `One(L): RngPadRes -> RngPadResElt` ### `One(L): RngPadResExt -> RngPadResExtElt` ### `One(L): FldPad -> FldPadElt` Given a local ring or field $L$, create the multiplicative identity of $L$. Note that if $L$ is an unbounded precision structure, this will only be the one element to the default precision of the ring, and hence only an approximation to the multiplicative identity of $L$. ### `Random(L): RngPad -> RngPadElt` ### `Random(L): RngPadRes -> RngPadResElt` ### `Random(L): RngPadResExt -> RngPadResExtElt` Given a local ring or field $L$, return a random element of $L$, which must be a quotient ring or bounded precision ring. The element will have the default precision of the ring. ### `Representative(L): RngPad -> RngPadElt` ### `Representative(L): RngPadRes -> RngPadResElt` ### `Representative(L): RngPadResExt -> RngPadResExtElt` ### `Representative(L): FldPad -> FldPadElt` Return an element of the local ring or field $L$. ### `elt: RngPad, RngElt -> RngPadElt` ### `elt: RngPad, [RngElt] -> RngPadElt` ### `L ! u: RngPad, RngElt -> RngPadElt` ### `L ! u: RngPad, [RngElt] -> RngPadElt` ### `elt: FldPad, RngElt -> FldPadElt` ### `elt: FldPad, [RngElt] -> FldPadElt` ### `L ! u: FldPad, RngElt -> FldPadElt` ### `L ! u: FldPad, [RngElt] -> FldPadElt` Coerce the object $u$ into the local ring or field $L$. The resulting element will have as much precision as possible. The element $u$ is allowed to be one of the following: **(i)** An integer. **(ii)** An element of ${\mathbb{Z}}/ p^m{\mathbb{Z}}$; (where $m$ is a precision). **(iii)** An element of the residue class field of $L$. **(iv)** An element of a local ring or field with something in common with $L$. **(v)** A rational number. If $L$ is a ring then $u$ must not have valuation in the denominator. **(vi)** An element of a valuation ring over the rationals with the same prime as $L$. **(vii)** A sequence $s$. In this case, the sequence $s$ is coerced to a sequence $t$ over the base ring or field of $L$. This sequence is coerced to $\sum_{i=1}^{\#t} t[i] L.1^{i-1}$. ### `elt: RngPad, RngElt, RngIntElt -> RngPadElt` ### `elt: RngPad, [RngElt], RngIntElt -> RngPadElt` ### `elt: FldPad, RngElt, RngIntElt -> FldPadElt` ### `elt: FldPad, [RngElt], RngIntElt -> FldPadElt` Create an element of the local ring or field $L$ by coercing $u$ into $L$ and returning with it precision $r$. ### `elt: RngPad, RngIntElt, RngElt, RngIntElt -> RngPadElt` ### `elt: RngPad, RngIntElt, [RngElt], RngIntElt -> RngPadElt` ### `elt: FldPad, RngIntElt, RngElt, RngIntElt -> FldPadElt` ### `elt: FldPad, RngIntElt, [RngElt], RngIntElt -> FldPadElt` Create an element of the local ring or field $L$ by coercing $u$ into $L$, multiplying by the $v$-th power of the uniformizing element and returning it with precision $r$. ### `BigO(x): RngPadElt -> RngPadElt` ### `BigO(x): FldPadElt -> FldPadElt` ### `O(x): RngPadElt -> RngPadElt` ### `O(x): FldPadElt -> FldPadElt` For an element $x$ of a local ring or field $L$ of valuation $v$, create an element of valuation $v$ and relative precision $0$. For rings this is the zero element in the quotient $L / \pi^v L$. ### `UniformizingElement(L): RngPad -> RngPadElt` ### `UniformizingElement(L): RngPadRes -> RndPadResElt` ### `UniformizingElement(L): RngPadResExt -> RndPadResExtElt` ### `UniformizingElement(L): FldPad -> FldPadElt` Given a local ring or field $L$, return the uniformizing element of $L$ to the default precision of $L$. ### `Example: eltcons (ex-9e5bd3)` Here we illustrate the usage of element constructors for local fields and imprecise zeros. ```magma > Zp := pAdicRing(5, 20); > I := UnramifiedExtension(Zp, 3); > R := PolynomialRing(I); > L := ext; > K := ext; > K; Totally ramified extension defined by the polynomial x^2 + 5 over Unramified extension defined by the polynomial x^3 + 3*x + 3 over 5-adic field mod 5^20 > elt; 64 + O(pi^40) > P := PrimeField(K); > elt; pi^2*7 + O(pi^6) > 4*$1; pi^2*3 + O(pi^6) > K!3/5; pi^-2*3 + O(pi^38) > O(K!40^200); O(pi^400) > Precision($1); 0 > O(K!0); O(pi^40) > O(K!1); O(1) > R := PolynomialRing(Integers()); > K := ext; > elt; 64 + O(pi^200) > Precision($1); 200 > K!3/10; -pi^-2*3944304526105059027058642826413931148366032175545115023851394653320311 + O(pi^198) ``` ### `Example: Eltcons Seq Weird (ex-b0924e)` There is a subtle interplay between the default precision of rings and fields and coercion of sequences, which we demonstrate here. We construct the 5-adic field $P$, and an unramified extension of $R$. We set the default precision of $R$ to be higher than that of $P$. ```magma > P := pAdicRing(5); > R := ext

; > P`DefaultPrecision; 20 > R`DefaultPrecision := 40; > x := Random(R); > x; -1579801843431963201369145587*R.1 - 680575730458975039033394769 + O(5^40) > s := [-680575730458975039033394769, -1579801843431963201369145587]; > R!s; -13585890629962*R.1 + 47482939261481 + O(5^20) > P`DefaultPrecision := 40; > R!s; -1579801843431963201369145587*R.1 - 680575730458975039033394769 + O(5^40) ``` As can be seen from the above, it is the default precision of the base ring, not the ring itself, which determines the precision of elements created by sequences. ## Element Decomposers ### `ElementToSequence(x): RngPadElt -> [ RngElt ]` ### `ElementToSequence(x): RngPadResElt -> [ RngElt ]` ### `ElementToSequence(x): RngPadResExtElt -> [ RngElt ]` ### `ElementToSequence(x): FldPadElt -> [ RngElt ]` ### `Eltseq(x): RngPadElt -> [ RngElt ]` ### `Eltseq(x): RngPadResElt -> [ RngElt ]` ### `Eltseq(x): RngPadResExtElt -> [ RngElt ]` ### `Eltseq(x): FldPadElt -> [ RngElt ]` ### `Coefficients(x): RngPadElt -> [ RngElt ]` ### `Coefficients(x): RngPadResElt -> [ RngElt ]` ### `Coefficients(x): RngPadResExtElt -> [ RngElt ]` ### `Coefficients(x): FldPadElt -> [ RngElt ]` Given an element $x$ of a degree $n$ extension $K$ of $L$, these functions return a sequence $s$ of elements of $L$ such that $x = \sum_{i=1}^{n} s[i] K.1^{i-1}$. ### `Coefficient(x, i): RngPadElt, RngIntElt -> RngPadElt` ### `Coefficient(x, i): RngPadResElt, RngIntElt -> RngPadResElt` ### `Coefficient(x, i): RngPadResExtElt, RngIntElt -> RngElt` ### `Coefficient(x, i): FldPadElt, RngIntElt -> FldPadElt` Equivalent to, but more efficient than, `Coefficients(x)[i]`. ### `Example: Gal Desc (ex-716c79)` We want to perform a Galois descent for a polynomial, i.e. we interpret the product of the Galois conjugates of a polynomial in a subring. ```magma > p := 3; > L := UnramifiedExtension(pAdicRing(p), 4 : Cyclotomic); > R := PolynomialRing(L); > g := x^2 + (a+a^-2)*x + (a^-1+a^3+1); > g; (1 + O(3^20))*x^2 + (-151999392*a^3 - 428033534*a^2 + 1509587217*a - 64512399 + O(3^20))*x + 2*a^3 + 307453058*a^2 - 1732356354*a - 151999391 + O(3^20) > a2 := a^(p^2); > g2 := R ! [ &+[Eltseq(c)[i]*a2^(i-1) : i in [1..4]] : c in Eltseq(g) ]; ``` Here `Eltseq` is being used to replace occurrences of $a$ in the element by $a_2$. ```magma > h := g * g2; > h; (1 + O(3^20))*x^4 + (774759822*a^3 - 1559939781*a^2 + 644136111*a + 143311364 + O(3^20))*x^3 + (73899384*a^3 - 333497478*a^2 + 1655979363*a + 158024680 + O(3^20))*x^2 + (989668189*a^3 - 661853000*a^2 - 1685887308*a + 122035547 + O(3^20))*x - 1630826887*a^3 - 328410694*a^2 - 175290219*a + 1601599448 + O(3^20) ``` The polynomial $g_2$ is the image of $g$ under the automorphism induced by the square of the Frobenius automorphism. The product $h = g g_2$ has coefficients in the unramified extension of degree 2, which is the fixed field under the square of the Frobenius and is generated by $a^{10}$. Next, we determine a representation for the polynomial $h$ with coefficients lying in this unramified extension of degree 2. ```magma > K := UnramifiedExtension(pAdicRing(p), 2 : Cyclotomic); > S := PolynomialRing(K); > M := RMatrixSpace(PrimeRing(L), 2, 4) ! 0; > V := RSpace(PrimeRing(L), 4); > M[1] := V ! Eltseq(a^0); > M[2] := V ! Eltseq(a^10); > sol := [ Solution(M, V ! Eltseq(c)) : c in Eltseq(h) ]; > h2 := S ! [ K ! Eltseq(s) : s in sol ]; > h2; (1 + O(3^20))*y^4 + (-1237301755*b + 505889265 + O(3^20))*y^3 + (-542504216*b + 1258167831 + O(3^20))*y^2 + (-280210015*b - 1205051127 + O(3^20))*y + 1213139407*b + 1602993886 + O(3^20) ```