// Numbas version: exam_results_page_options {"name": "Der K\u00f6rper $\\mathbb F_4$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Der K\u00f6rper $\\mathbb F_4$", "tags": [], "metadata": {"description": "

Deduce some properties of elements of $\\mathbb F_4$, given its multiplication table. (E.g., which element is the zero, the one element, ...)

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Es sei $K = \\{a, b, c, d\\}$ ein Körper mit $4$ Elementen und der folgenden Multiplikationstabelle:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\cdot$	$a$	$b$	$c$	$d$
$a$	$\\var{latex(m[0][0])}$	$\\var{latex(m[0][1])}$	$\\var{latex(m[0][2])}$	$\\var{latex(m[0][3])}$
$b$	$\\var{latex(m[1][0])}$	$\\var{latex(m[1][1])}$	$\\var{latex(m[1][2])}$	$\\var{latex(m[1][3])}$
$c$	$\\var{latex(m[2][0])}$	$\\var{latex(m[2][1])}$	$\\var{latex(m[2][2])}$	$\\var{latex(m[2][3])}$
$d$	$\\var{latex(m[3][0])}$	$\\var{latex(m[3][1])}$	$\\var{latex(m[3][2])}$	$\\var{latex(m[3][3])}$

Als Antwort auf die Fragen unten wird immer einer der Buchstaben a, b, c, d erwartet.

", "advice": "

Es gilt $0\\cdot x = x\\cdot 0 = 0$ für alle $x\\in K$, also ist das Nullelement das (eindeutig bestimmte) Element, in dessen Zeile und Spalte alle Einträge mit diesem Element übereinstimmen.

Es gilt $1\\cdot x = x\\cdot 1 = x$ für alle $x\\in K$, also ist das Einselement das (eindeutig bestimmte) Element, in dessen Zeile und Spalte die Einträge $a$, $b$, $c$, $d$ (in dieser Reihenfolge) lauten.

In jedem Körper gilt $(-1)^2 = 1$, also muss in der Multiplikationstabelle der Diagonaleintrag bei $-1$ gleich $\\var{latex(elt_to_letter[1])}$ sein - denn dies ist das Element $1$ in $K$. Das kommt aber nur ein einziges Mal vor, nämlich bei $\\var{latex(elt_to_letter[1])}$ selbst. Es gilt also $-1=\\var{latex(elt_to_letter[1])}$ in $K$. Wir können das umformulieren zu $-1=1$ in $K$, also folgt insbesondere $1+1=0$.

Der Ausdruck $\\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[1])}$ kann weder den Wert $\\var{latex(elt_to_letter[2])}$ noch den Wert $\\var{latex(elt_to_letter[1])}$ haben, weil sonst das andere Element das Nullelement wäre; das Nullelement ist aber $\\var{latex(elt_to_letter[0])}$. Der Wert kann auch nicht $\\var{latex(elt_to_letter[0])}$ sein, denn das würde bedeuten, dass $\\var{latex(elt_to_letter[2])} = -\\var{latex(elt_to_letter[1])}$ ist, was im Widerspruch zu Teil b) steht. Also gilt $\\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[1])} = \\var{latex(elt_to_letter[3])}$. Für den zweiten Teil kann man ähnlich argumentieren, oder den ersten Teil benutzen und auf beiden Seiten $\\var{latex(elt_to_letter[1])}$ abziehen.

Hier kann man zum Beispiel Teil c) und Teil b) anwenden:

\\[\\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[3])} = \\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[1])} = (1+1)\\cdot \\var{latex(elt_to_letter[2])} +\\var{latex(elt_to_letter[1])} = \\var{latex(elt_to_letter[1])} (= 1).\\]

", "rulesets": {}, "extensions": [], "variables": {"sigma": {"name": "sigma", "group": "Ungrouped variables", "definition": "shuffle([0, 1, 2, 3])", "description": "

The permutation that shuffles the elements of $\\mathbb F_4$.

", "templateType": "anything"}, "elt_to_letter": {"name": "elt_to_letter", "group": "Ungrouped variables", "definition": "map([\"a\", \"b\", \"c\", \"d\"][x], x, sigma1)", "description": "", "templateType": "anything"}, "mult": {"name": "mult", "group": "Ungrouped variables", "definition": "[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 3, 1], [0, 3, 1, 2]]", "description": "

Multiplication table of $\\mathbb F_4$

", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "map(map(elt_to_letter[x], x, map(mult[sigma[y]][z], z, sigma)), y, 0..3)", "description": "", "templateType": "anything"}, "sigma1": {"name": "sigma1", "group": "Ungrouped variables", "definition": "map(indices(sigma, x)[0], x, 0..3)", "description": "

$\\sigma^{-1}$

", "templateType": "anything"}}, "variablesTest": {"condition": "!(sigma[0] = 0)", "maxRuns": 100}, "ungrouped_variables": ["sigma", "elt_to_letter", "mult", "m", "sigma1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Geben Sie das neutrale Element $0$ bezüglich der Addition und das neutrale Element $1$ bezüglich der Multiplikation an:

$0_K=$ [[0]]

$1_K=$ [[1]]

", "gaps": [{"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[0]}", "displayAnswer": "", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[1]}", "displayAnswer": "", "matchMode": "exact"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Berechnen Sie in $K$:

$-1_K = $ [[0]]

$\\var{latex(elt_to_letter[1])} + \\var{latex(elt_to_letter[1])} =$ [[1]]

", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Für den ersten Teil: Überlegen Sie, was $(-1)^2$ ist.

Für den zweiten Teil können Sie den ersten Teil anwenden.

"}], "gaps": [{"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[1]}", "displayAnswer": "", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[0]}", "displayAnswer": "", "matchMode": "exact"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Berechnen Sie in $K$:

$\\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[1])} =$ [[0]]

$\\var{latex(elt_to_letter[3])} + \\var{latex(elt_to_letter[1])} =$ [[1]]

Eine Möglichkeit ist, mit dem Ausschlussverfahren zu arbeiten. Zum Beispiel: $x+y = y \\Rightarrow x = 0$ (man kann $y$ auf beiden Seiten der ersten Gleichung abziehen), $x+y = x \\Rightarrow \\dots$.

"}], "gaps": [{"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[3]}", "displayAnswer": "", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{elt_to_letter[2]}", "displayAnswer": "", "matchMode": "exact"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Berechnen Sie in $K$:

$\\var{latex(elt_to_letter[2])} + \\var{latex(elt_to_letter[3])} = $ [[0]]